// Numbas version: exam_results_page_options
{"name": "Max and Min 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"third1": {"name": "third1", "group": "Ungrouped variables", "definition": "16*(k-b)", "description": "", "templateType": "anything"}, "l": {"name": "l", "group": "Ungrouped variables", "definition": "k-random(1..3)", "description": "", "templateType": "anything"}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "a+random(1..6)", "description": "", "templateType": "anything"}, "gma": {"name": "gma", "group": "Ungrouped variables", "definition": "(xma-a)*(xma-b)^3", "description": "", "templateType": "anything"}, "s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(3,6)", "description": "", "templateType": "anything"}, "third2": {"name": "third2", "group": "Ungrouped variables", "definition": "8*(b-k)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "4*k-3*a", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything"}, "valbegin": {"name": "valbegin", "group": "Ungrouped variables", "definition": "(l-a)*(l-b)^3", "description": "", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "b+random(1..3)", "description": "", "templateType": "anything"}, "gmi": {"name": "gmi", "group": "Ungrouped variables", "definition": "(k-a)*(k-b)^3", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "if(c=3,random(-9..9),random(-9..9#2))", "description": "", "templateType": "anything"}, "xma": {"name": "xma", "group": "Ungrouped variables", "definition": "if(valend > valbegin,m,l)", "description": "", "templateType": "anything"}, "valend": {"name": "valend", "group": "Ungrouped variables", "definition": "(m-a)*(m-b)^3", "description": "", "templateType": "anything"}}, "advice": "\n  <h4>First derivative</h4>\n    <p>Differentiating we have:</p>\n    <p>\\[\\begin{eqnarray*} g'(x)&amp;=&amp;\\simplify{(x-{b})^3+3*(x-{a})*(x-{b})^2}\\\\ &amp;=&amp;\\simplify{(x-{b})^2(3*(x-{a})+x-{b})}\\\\ &amp;=&amp;\\simplify{4*(x-{k})*(x-{b})^2} \\end{eqnarray*} \\] and we have factorised the expression.</p>\n    <h4>Stationary points</h4>\n    <p>These are given by solving $g'(x)=0 \\Rightarrow x=\\var{k},\\;\\;\\mbox{or }x=\\var{b}$</p>\n    <p>Therefore the least stationary point is $x=\\var{k}$ and the greatest is $x=\\var{b}$ and we see that both stationary points are in $I$.</p>\n    <h4>Second derivative.</h4>\n    <p>The second derivative is given by:<br/>\\[\\begin{eqnarray*} g''(x)&amp;=&amp;\\simplify{4*(x-{b})^2+8*(x-{k})(x-{b})}\\\\ &amp;=&amp;\\simplify{4*(x-{b})(3*x-{2*k+b})} \\end{eqnarray*} \\]</p>\n    <h4>Local Minimum</h4>\n    <p>At the stationary point $x=\\var{k}$ we have $g''(\\var{k})=\\var{4*(k-b)^2} \\gt 0$.</p>\n    <p>Hence $x=\\var{k}$ is a local minimum.</p>\n    <h4>The other stationary point</h4>\n    <p>The value at $x=\\var{b}$ is $g(\\var{b})= 0$.</p>\n    <p>Hence this test fails at this point and we proceed to use the third derivative to see in more information can be gained.</p>\n    <h4>Third derivative</h4>\n    <p>We see that $g'''(x)=\\simplify{8*(3*x-{k+2*b})}$.</p>\n    <p>Testing the stationary point using the third derivative gives:</p>\n    <p>$g'''(\\var{b})=\\var{8*(b-k)} \\neq 0$.</p>\n    <p>Therefore there cannot be an extremum point at $x=\\var{b}$.</p>\n    <h4>Finding the global maximum and minimum on $I$</h4>\n    <p>First we find the values at the endpoints of the interval 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"absdiff", "useCustomName": false, "answerSimplification": "std", "marks": 1, "checkingAccuracy": 0.001, "notallowed": {"message": "<p>Factorise the expression</p>", "strings": ["x^2", "x^3"], "partialCredit": 0, "showStrings": false}, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "failureRate": 1, "vsetRangePoints": 5, "musthave": {"message": "<p>Factorise the expression</p>", "strings": ["(", ")"], "partialCredit": 0, "showStrings": false}, "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "valuegenerators": [{"name": "x", "value": ""}], "showPreview": true, "scripts": {}, "variableReplacements": [], "customName": "", "unitTests": [], "answer": "4*(x + {-k}) * (x + {-b})^2", "variableReplacementStrategy": "originalfirst"}], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "prompt": "\n      <p>Input the first derivative of $g$ here, factorised into a product of two factors in the form $g&#39;(x)=c(x-a)(x-b)^2$for suitable integers $a$, $b$  and $c$:</p>\n      \n      <p>$g&#39;(x)=\\;\\;$[[0]]</p>\n      \n      ", "unitTests": [], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true}, {"customMarkingAlgorithm": "", "useCustomName": false, "scripts": {}, "marks": 0, "variableReplacements": [], "gaps": [{"checkingType": "absdiff", "useCustomName": false, "answerSimplification": "std", "marks": 1, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "valuegenerators": [], "showPreview": true, "scripts": {}, "variableReplacements": [], "customName": "", "unitTests": [], "answer": "{k}", "variableReplacementStrategy": "originalfirst"}, {"checkingType": "absdiff", "useCustomName": false, "answerSimplification": "std", "marks": 1, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "valuegenerators": [], "showPreview": true, "scripts": {}, "variableReplacements": [], "customName": "", "unitTests": [], "answer": "{b}", "variableReplacementStrategy": "originalfirst"}, {"displayColumns": 0, "customMarkingAlgorithm": "", "matrix": [1, 0], "choices": ["<p>Yes</p>", "<p>No</p>"], "useCustomName": false, "scripts": {}, "marks": 0, "variableReplacements": [], "maxMarks": 0, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "showCellAnswerState": true, "unitTests": [], "shuffleChoices": true, "minMarks": 0, "distractors": ["", ""], "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "showFeedbackIcon": true}], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "prompt": "\n      <h4>Find the stationary points of $g$, here thought of as $g:\\mathbb{R} \\rightarrow \\mathbb{R}$.</h4>\n      \n      <p>Least stationary point: [[0]]</p>\n      \n      <p>Greatest stationary point: [[1]]</p>\n      \n      <p>Do both these stationary points lie in the interval $I$ ? [[2]]</p>\n      \n      ", "unitTests": [], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true}, {"customMarkingAlgorithm": "", "useCustomName": false, "scripts": {}, "marks": 0, "variableReplacements": [], "gaps": [{"checkingType": "absdiff", "useCustomName": false, "answerSimplification": "std", "marks": 1, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "valuegenerators": [{"name": "x", "value": ""}], "showPreview": true, "scripts": {}, "variableReplacements": [], "customName": "", "unitTests": [], "answer": "4*(x-{b})*(3*x-{2*k+b})", "variableReplacementStrategy": "originalfirst"}, {"displayColumns": 0, "customMarkingAlgorithm": "", "matrix": [1, 0, 0], "choices": ["<p>A local minimum.</p>", "<p>A local maximum.</p>", "<p>Uncertain as the second derivative test fails.</p>"], "useCustomName": false, "scripts": {}, "marks": 0, "variableReplacements": [], "maxMarks": 0, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "showCellAnswerState": true, "unitTests": [], "shuffleChoices": true, "minMarks": 0, "distractors": ["", "", ""], "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "showFeedbackIcon": true}, {"displayColumns": 0, "customMarkingAlgorithm": "", "matrix": [0, 0, 1], "choices": ["<p>A local minimum.</p>", "<p>A local maximum.</p>", "<p>Uncertain as the second derivative test fails.</p>"], "useCustomName": false, "scripts": {}, "marks": 0, "variableReplacements": [], "maxMarks": 0, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "showCellAnswerState": true, "unitTests": [], "shuffleChoices": true, "minMarks": 0, "distractors": ["", "", ""], "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "showFeedbackIcon": true}], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "prompt": "\n      <p>Input the second derivative of $g$:</p>\n      \n      <p>$g&#39;&#39;(x)=\\;\\;$ [[0]]</p>\n      \n      <p>Using $g&#39;&#39;(x)$, determine more information about the stationary points:</p>\n      \n      <p>Least stationary point is: (Choose one of the following)<br/>[[1]]</p>\n      \n      <p>Greatest stationary point is: (Choose one of the following)<br/>[[2]]</p>\n      \n      ", "unitTests": [], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true}, {"customMarkingAlgorithm": "", "useCustomName": false, "scripts": {}, "marks": 0, "variableReplacements": [], "gaps": [{"checkingType": "absdiff", "useCustomName": false, "answerSimplification": "std", "marks": 1, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "valuegenerators": [{"name": "x", "value": ""}], "showPreview": true, "scripts": {}, "variableReplacements": [], "customName": "", "unitTests": [], "answer": "8*(3*x-{k+2*b})", "variableReplacementStrategy": "originalfirst"}, {"correctAnswerFraction": false, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "useCustomName": false, "scripts": {}, "marks": 0.5, "variableReplacements": [], "allowFractions": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "mustBeReduced": false, "unitTests": [], "maxValue": "{third1}", "showFractionHint": true, "minValue": "{third1}", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true}, {"correctAnswerFraction": false, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "useCustomName": false, "scripts": {}, "marks": 0.5, "variableReplacements": [], "allowFractions": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "mustBeReduced": false, "unitTests": [], "maxValue": "{third2}", "showFractionHint": true, "minValue": "{third2}", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true}, {"displayColumns": 0, "customMarkingAlgorithm": "", "matrix": [0, 1, 0, 0, 0], "choices": ["<p>{k} is not a local minimum or a local maximum.</p>", "<p>{b} is not a local minimum or a local maximum.</p>", "<p>{b} is not a local minimum or a local maximum and neither is {k}.</p>", "<p>{k} is a local maximum and {b} is a local minimum.</p>", "<p>{k} is a local minimum and {b} is a local maximum.</p>"], "useCustomName": false, "scripts": {}, "marks": 0, "variableReplacements": [], "maxMarks": 0, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "showCellAnswerState": true, "unitTests": [], "shuffleChoices": true, "minMarks": 0, "distractors": ["", "", "", "", ""], "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "showFeedbackIcon": true}], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "prompt": "\n      <p>Using the third derivative answer the following questions:<br/>$g&#39;&#39;&#39;(x) = \\;\\;$[[0]]</p>\n      \n      <p>If $a$ is the least stationary point then $g&#39;&#39;&#39;(a) =\\;\\;$[[1]]</p>\n      \n      <p>If $b$ is the other stationary point then $g&#39;&#39;&#39;(b) =\\;\\;$[[2]]</p>\n      \n      <p>This information tells us that: (Choose one of the following).<br/>[[3]]</p>\n      \n      ", "unitTests": [], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true}, {"customMarkingAlgorithm": "", "useCustomName": false, "scripts": {}, "marks": 0, "variableReplacements": [], "gaps": [{"correctAnswerFraction": false, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "useCustomName": false, "scripts": {}, "marks": 0.5, "variableReplacements": [], "allowFractions": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "mustBeReduced": false, "unitTests": [], "maxValue": "{valbegin}", "showFractionHint": true, "minValue": "{valbegin}", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true}, {"correctAnswerFraction": false, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "useCustomName": false, "scripts": {}, "marks": 0.5, "variableReplacements": [], "allowFractions": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "mustBeReduced": false, "unitTests": [], "maxValue": "{valend}", "showFractionHint": true, "minValue": "{valend}", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true}], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "prompt": "\n      <p>What are the following values at the end points of the interval $I$ ?</p>\n      \n      <p>$g(\\var{l})=\\;\\;$ [[0]]</p>\n      \n      <p>$g(\\var{m})=\\;\\;$ [[1]]</p>\n      \n      <p>Input both to 2 decimal places.</p>\n      \n      ", "unitTests": [], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true}, {"customMarkingAlgorithm": "", "useCustomName": false, "scripts": {}, "marks": 0, "variableReplacements": [], "gaps": [{"correctAnswerFraction": false, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "useCustomName": false, "scripts": {}, "marks": 1, "variableReplacements": [], "allowFractions": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "mustBeReduced": false, "unitTests": [], "maxValue": "{xma}", "showFractionHint": true, "minValue": "{xma}", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true}, {"correctAnswerFraction": false, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "useCustomName": false, "scripts": {}, "marks": 1, "variableReplacements": [], "allowFractions": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "mustBeReduced": false, "unitTests": [], "maxValue": "{gma}", "showFractionHint": true, "minValue": "{gma}", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true}, {"correctAnswerFraction": false, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "useCustomName": false, "scripts": {}, "marks": 1, "variableReplacements": [], "allowFractions": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "mustBeReduced": false, "unitTests": [], "maxValue": "{k}", "showFractionHint": true, "minValue": "{k}", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true}, {"correctAnswerFraction": false, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "useCustomName": false, "scripts": {}, "marks": 1, "variableReplacements": [], "allowFractions": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "mustBeReduced": false, "unitTests": [], "maxValue": "{gmi}", "showFractionHint": true, "minValue": "{gmi}", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true}], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "prompt": "\n      <h4>Global Maximum</h4>\n      \n      <p>At what value of $x \\in I$ does $g$ have a global maximum ?</p>\n      \n      <p>$x=\\;\\;$ [[0]]</p>\n      \n      <p>Value of $g$ at this global maximum = [[1]].</p>\n      \n      <h4>Global Minimum</h4>\n      \n      <p>At what value of $x \\in I$ does $g$ have a global minimum ?</p>\n      \n      <p>$x=\\;\\;$ [[2]]</p>\n      \n      <p>Value of $g$ at this global minimum = [[3]].</p>\n      \n      ", "unitTests": [], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true}], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variable_groups": [], "statement": "\n  <p>Let $I=[\\var{l},\\var{m}]$ be an interval and let $g: I \\rightarrow \\mathbb{R}$ be a function defined on this interval<br/>given by :\\[g(x) = \\simplify{(x-{a})*(x-{b})^3}\\]</p>\n  \n  ", "metadata": {"description": "<p>$I$ compact interval, $g:I\\rightarrow I$, $g(x)=(x-a)(x-b)^2$. Stationary points in interval. Find local and global maxima and minima of $g$ on $I$.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "preamble": {"css": "", "js": ""}, "functions": {}, "tags": [], "ungrouped_variables": ["third1", "xma", "valbegin", "gmi", "d", "s1", "k", "gma", "b", "third2", "valend", "c", "m", "l", "a"], "contributors": [{"name": "Paul Howes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/632/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Paul Howes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/632/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}