// Numbas version: exam_results_page_options
{"name": "Max and Min v2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Max and Min v2", "tags": [], "metadata": {"description": "<p>$I$ compact interval, $g:I\\rightarrow I,\\;g(x)=ax^3+bx^2+cx+d$. Find stationary points, local and global maxima and minima of $g$ on $I$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\n    <p>Let $I=[\\var{l},\\var{m}]$ be an interval and let $g: I \\rightarrow I$ be a function defined on this interval<br/>given by :\\[g(x) = \\simplify{{c}/3*x^3+ {-c*(a+b)}/2*x^2+{c*a*b}*x+{d}}\\]</p>\n    \n  ", "advice": "<p>Differentiating, we have:</p>\n<p>\\[g'(x)=\\simplify{{c}*x^2+{-c*(a+b)}*x+{c*a*b}={c}*(x+{-a})(x-{b})}\\]</p>\n<p>Note that we have already factorised the derivative.</p>\n<p>Stationary points are given by solving $g'(x)=0 \\Rightarrow x=\\var{a},\\;\\;\\mbox{or }x=\\var{b}$</p>\n<p>So the least stationary point is $x=\\var{a}$ and the greatest is $x=\\var{b}$.</p>\n<p>Since $\\var{a} &gt; \\var{l}$ and $\\var{b} \\lt \\var{m}$ we have that both stationary points are in $I$.</p>\n<p>The second derivative is given by \\[g''(x)=\\simplify{{2*c}*x-{c*(a+b)}}\\]</p>\n<h4>Local Maximum</h4>\n<p>At the stationary point $x=\\var{a}$ we have $g''(\\var{a})=\\var{c*a-c*b} \\lt 0$.</p>\n<p>Hence at this value of $x$ we have a local maximum.</p>\n<p>The value of the function $g$ at this local maximum is $g(\\var{a})= \\var{valmax}$.</p>\n<h4>Local Minimum</h4>\n<p>At the stationary point $x=\\var{b}$ we have $g''(\\var{b})=\\var{c*b-c*a} \\gt 0$.</p>\n<p>Hence this point is a local minimum.</p>\n<p>The value of the function $g$ at this local minimum is $g(\\var{b})= \\var{valmin}$.</p>\n<h4>Global Maximum</h4>\n<p>First we find the values at the endpoints of the interval $I=[\\var{l},\\var{m}]$ are:</p>\n<p>$g(\\var{l})=\\var{valbegin}$ to 3 decimal places.</p>\n<p>$g(\\var{m})=\\var{valend}$ to 3 decimal places.</p>\n<p>To find the global maximum note that we are only concerned with the values of $g$ on the interval $I$.</p>\n<p>So we proceed by comparing the values of the function at the endpoints with the local maximum.</p>\n<p><strong>a)</strong> If the value at the local maximum is greater than either of the values at the endpoints then this is the global maximum on the interval.</p>\n<p><strong>b)</strong> Otherwise if the greatest value of the function at the endpoints is greater than the local maximum then this is the global maximum.</p>\n<p>\\[\\begin{array}{c|c|c|c} x &amp; \\mbox{Local Maximum}=\\var{a} &amp; \\var{l} \\in I &amp; \\var{m} \\in I \\\\ \\hline\\\\ g(x)&amp; \\var{valmax} &amp; \\var{valbegin} &amp; \\var{valend} \\\\ \\end{array} \\]</p>\n<p>So for our example we see that the global maximum occurs at $x=\\var{gma}$ and $g(\\var{gma})=\\var{valgmax}$.</p>\n<h4>Global Minimum</h4>\n<p>We proceed as for the global maximum by comparing the values of the function at the endpoints with the local minimum.</p>\n<p><strong>a)</strong> If the value at the local minimum is less than either of the values at the endpoints then this is the global minimum on the interval.</p>\n<p><strong>b)</strong> Otherwise if the least value of the function at the endpoints is less than the local minimum then this is the global minimum.</p>\n<p>\\[\\begin{array}{c|c|c|c} x &amp; \\mbox{Local Minimum}=\\var{b} &amp; \\var{l} \\in I &amp; \\var{m} \\in I \\\\ \\hline\\\\ g(x)&amp; \\var{valmin} &amp; \\var{valbegin} &amp; \\var{valend} \\\\ \\end{array} \\]</p>\n<p>In our example we see that the global minimum occurs at $x=\\var{gmi}$ and $g(\\var{gmi})=\\var{valgmin}$.</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "extensions": [], "variables": {"valmin": {"name": "valmin", "group": "Ungrouped variables", "definition": "round(rawvalmin)", "description": "", "templateType": "anything"}, "valgmax": {"name": "valgmax", "group": "Ungrouped variables", "definition": "max(valend,valmax)", "description": "", "templateType": "anything"}, "rawvalend": {"name": "rawvalend", "group": "Ungrouped variables", "definition": "(c/3) * m^ 3 + (-c * (a + b)) / 2 * m^ 2 + (c * a * b) * m+ d", "description": "", "templateType": "anything"}, "gmi": {"name": "gmi", "group": "Ungrouped variables", "definition": "if(valbegin<valmin,l,b)", "description": "", "templateType": "anything"}, "gma": {"name": "gma", "group": "Ungrouped variables", "definition": "if(valend>valmax,m,a)", "description": "", "templateType": "anything"}, "l": {"name": "l", "group": "Ungrouped variables", "definition": "if(valmin=temp1,l1-0.5,l1)", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "if(c=3,random(-9..9),random(-9..9#2))", "description": "", "templateType": "anything"}, "m1": {"name": "m1", "group": "Ungrouped variables", "definition": "b+random(0.5..3#0.5)", "description": "", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "if(valmax=temp2,m1+0.5,m1)", "description": "", "templateType": "anything"}, "temp2": {"name": "temp2", "group": "Ungrouped variables", "definition": "precround(rtemp2,2)", "description": "", "templateType": "anything"}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "rawvalmax": {"name": "rawvalmax", "group": "Ungrouped variables", "definition": "(c/3) * a^ 3 + (-c * (a + b)) / 2 * a^ 2 + (c * a * b) * a + d", "description": "", "templateType": "anything"}, "valend": {"name": "valend", "group": "Ungrouped variables", "definition": "precround(rawvalend,2)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "s*random(1..5)", "description": "", "templateType": "anything"}, "valmax": {"name": "valmax", "group": "Ungrouped variables", "definition": "round(rawvalmax)", "description": "", "templateType": "anything"}, "temp1": {"name": "temp1", "group": "Ungrouped variables", "definition": "precround(rtemp1,2)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "if(c=6,a+random(1..9),random(2..10#2)+a)", "description": "", "templateType": "anything"}, "rawvalmin": {"name": "rawvalmin", "group": "Ungrouped variables", "definition": "(c/3) * b^ 3 + (-c * (a + b)) / 2 * b^ 2 + (c * a * b) * b + d", "description": "", "templateType": "anything"}, "valgmin": {"name": "valgmin", "group": "Ungrouped variables", "definition": "min(valbegin,valmin)", "description": "", "templateType": "anything"}, "rawvalbegin": {"name": "rawvalbegin", "group": "Ungrouped variables", "definition": "(c/3) * l^ 3 + (-c * (a + b)) / 2 * l^ 2 + (c * a * b) * l+ d", "description": "", "templateType": "anything"}, "l1": {"name": "l1", "group": "Ungrouped variables", "definition": "a-random(0.5..3#0.5)", "description": "", "templateType": "anything"}, "valbegin": {"name": "valbegin", "group": "Ungrouped variables", "definition": "precround(rawvalbegin,2)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(3,6)", "description": "", "templateType": "anything"}, "rtemp1": {"name": "rtemp1", "group": "Ungrouped variables", "definition": "(c/3) * l1^ 3 + (-c * (a + b)) / 2 * l1^ 2 + (c * a * b) * l1+ d", "description": "", "templateType": "anything"}, "rtemp2": {"name": "rtemp2", "group": "Ungrouped variables", "definition": "(c/3) * m1^ 3 + (-c * (a + b)) / 2 * m1^ 2 + (c * a * b) * m1+ d", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["temp2", "temp1", "gmi", "valmin", "gma", "s", "m1", "valbegin", "a", "valmax", "valgmax", "valgmin", "rawvalend", "rawvalmax", "b", "rawvalbegin", "rawvalmin", "rtemp2", "c", "rtemp1", "d", "m", "l", "valend", "l1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n        <p>Input the first derivative of $g$ here, factorised into a product of two linear factors in the form $g&#39;(x)=c(x-a)(x-b)$for suitable integers $a$, $b$  and $c$:</p>\n        \n        <p>$g&#39;(x)=\\;\\;$[[0]]</p>\n        \n      ", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{c} * (x + {-a}) * (x + {-b})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "musthave": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "<p>Factorise the expression</p>"}, "notallowed": {"strings": ["^", "x*x", "xx", "x x"], "showStrings": false, "partialCredit": 0, "message": "<p>Factorise the expression</p>"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n      <h4>Find the stationary points of $g$.</h4>\n      <p>Least stationary point: [[0]] &nbsp; &nbsp;Greatest stationary point: [[1]]</p>\n      <p>Do both these stationary points lie in the interval $I$ ? [[2]]</p>\n      ", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a}", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{b}", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>Yes</p>", "<p>No</p>"], "matrix": [1, 0], "distractors": ["", ""]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n        <p>Input the second derivative of $g$:</p>\n        \n        <p>$g&#39;&#39;(x)=\\;\\;$ [[0]]</p>\n        \n        <p>Hence find all local maxima and minima given by the stationary points</p>\n        \n        <p>Local maximum is at $x=\\;\\;$ [[1]] and the value of the function at the local maximum = [[2]]</p>\n        \n        <p>Local minimum is at $x=\\;\\;$ [[3]] and the value of the function at the local minimum = [[4]]</p>\n        \n      ", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(({(2 * c)} * x) + ( - {(c * (a + b))}))", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a}", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{valmax}", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{b}", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{valmin}", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What are the following values at the end points of the interval $I$ ?</p>\n<p>$g(\\var{l})=\\;\\;$ [[0]] &nbsp; &nbsp; &nbsp;$g(\\var{m})=\\;\\;$ [[1]]</p>\n<p>Input both to 2 decimal places.</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "valbegin", "maxValue": "valbegin", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "valend", "maxValue": "valend", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<h4>Global Maximum</h4>\n<p>At what value of $x \\in I$ does $g$ have a global maximum ?</p>\n<p>$x=\\;\\;$ [[0]]</p>\n<p>Value of $g$ at this global maximum = [[1]] (input to 2 decimal places).</p>\n<h4>Global Minimum</h4>\n<p>At what value of $x \\in I$ does $g$ have a global minimum ?</p>\n<p>$x=\\;\\;$ [[2]]</p>\n<p>Value of $g$ at this global minimum = [[3]] (input to 2 decimal places).</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{gma}", "maxValue": "{gma}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "valgmax", "maxValue": "valgmax", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{gmi}", "maxValue": "{gmi}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "valgmin", "maxValue": "valgmin", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Daniel Mansfield", "profile_url": 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