// Numbas version: exam_results_page_options
{"name": "Paul 's copy of Max and Min 5", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"b": {"group": "Ungrouped variables", "definition": "random(1..9)", "name": "b", "templateType": "anything", "description": ""}, "d": {"group": "Ungrouped variables", "definition": "b+random(1..3)", "name": "d", "templateType": "anything", "description": ""}, "valbegin": {"group": "Ungrouped variables", "definition": "precround(a*c/(c^2+b^2),3)", "name": "valbegin", "templateType": "anything", "description": ""}, "lmi": {"group": "Ungrouped variables", "definition": "if(a<0,b,-b)", "name": "lmi", "templateType": "anything", "description": ""}, "valend": {"group": "Ungrouped variables", "definition": "precround(a*d/(d^2+b^2),3)", "name": "valend", "templateType": "anything", "description": ""}, "c": {"group": "Ungrouped variables", "definition": "-b-random(1..3)", "name": "c", "templateType": "anything", "description": ""}, "valmax": {"group": "Ungrouped variables", "definition": "-valmin", "name": "valmax", "templateType": "anything", "description": ""}, "tol": {"group": "Ungrouped variables", "definition": "0.001", "name": "tol", "templateType": "anything", "description": ""}, "lma": {"group": "Ungrouped variables", "definition": "if(a>0,b,-b)", "name": "lma", "templateType": "anything", "description": ""}, "s": {"group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s", "templateType": "anything", "description": ""}, "a": {"group": "Ungrouped variables", "definition": "s*random(1..9)", "name": "a", "templateType": "anything", "description": ""}, "valmin": {"group": "Ungrouped variables", "definition": "precround(-abs(a)*b/(2*b^2),3)", "name": "valmin", "templateType": "anything", "description": ""}}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>The function $g(x)$ is continuous and differentiable at all points in $\\mathbb{R}$.</p>\n<p>Using the quotient rule for differentiation we see that<br/>\\[\\begin{eqnarray*}g'(x)&amp;=&amp;\\simplify{({a}*(x^2+{b^2})-{2*a}*x^2)/(x^2+{b^2})^2}\\\\ &amp;=&amp;\\simplify{({-a}*(x-{b})(x+{b}))/(x^2+{b^2})^2} \\end{eqnarray*} \\]</p>\n<h4>Stationary Points.</h4>\n<p>The stationary points are given by solving $g'(x)=0$.</p>\n<p>$g'(x)=0 \\Rightarrow \\simplify{{-a}*(x-{b})(x+{b})=0} \\Rightarrow x=\\var{b} \\mbox{ or } x=\\var{-b}$</p>\n<p>We see that both stationary points are in the inerval $I$.</p>\n<p>The second derivative can be found by applying the quotient rule to the derivative of $g(x)$ and we obtain:</p>\n<p>Using the quotient rule for differentiation we see that<br/>\\[\\begin{eqnarray*}g''(x)&amp;=&amp;\\simplify[std]{({-2*a}*x*(x^2+{b^2})^2+{4*a}*x*(x^2-{b^2})(x^2+{b^2}))/(x^2+{b^2})^4}\\\\ &amp;=&amp;\\simplify[std]{({2*a}*x*(x^2-{3*b^2}))/(x^2+{b^2})^3} \\end{eqnarray*} \\]</p>\n<p>The nature of the stationary points are determined by evaluating $g''(x)$ at the stationary points.</p>\n<p>For $x= \\var{lma}$ we have: \\[g''(\\var{lma})= \\simplify[std]{-{abs(a)}/{2*b^3}} \\lt 0\\]</p>\n<p>Hence is a local maximum.</p>\n<p>Evaluating the function at $x=\\var{lma}$ gives $g(\\var{lma})=\\var{valmax}$ to 3 decimal places.</p>\n<p>For $x= \\var{lmi}$ we have: \\[g''(\\var{lmi})= \\simplify[std]{{abs(a)}/{2*b^3}} \\gt 0\\]</p>\n<p>Hence is a local minimum.</p>\n<p>Evaluating the function at $x=\\var{lmi}$ gives $g(\\var{lmi})=\\var{valmin}$ to 3 decimal places.</p>\n<p>The values of $g$ at the endpoints are:</p>\n<p>$g(\\var{c})=\\var{valbegin}$ and $g(\\var{d})=\\var{valend}$ to 3 decimal places.</p>\n<h4>Global Maximum and Minimum</h4>\n<p>Since $g$ has a finite limit of $0$ as $x \\rightarrow \\pm\\infty$ and we have that $0$ lies between the local minimum value $\\var{valmin}$ and the local maximum value $\\var{valmax}$ (and these occur at values in $I$).</p>\n<p>then:</p>\n<p><strong>Global Maximum</strong>: The local maximum of $g$ we have found at $x=\\var{lma} \\in I$ must be a global maximum and similarly,</p>\n<p><strong>Global Minimum</strong>: The local minimum of $g$ we have found at $x=\\var{lmi} \\in I$ must be a global minimum.</p>\n<p>So we have shown \\[\\forall x \\in \\mathbb{R},\\;\\;\\var{valmin} \\le g(x) \\le \\var{valmax}\\]</p>\n<p>(all to 3 decimal places).</p>", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>$I$ compact interval. $\\displaystyle g: I \\rightarrow I,&nbsp;g(x)=\\frac{ax}{x^2+b^2}$. Find stationary points and local maxima, minima. Using limits, has $g$ a global max, min?&nbsp;</p>"}, "tags": [], "functions": {}, "statement": "\n    <p>Let $I=[\\var{c},\\var{d}]$ be an interval and let $g: I \\rightarrow I$ be the function given by:<br/>\\[g(x)=\\simplify{{a}*x/(x^2+{b}^2)}\\]</p>\n    \n    <p>Answer the following questions. There are seven parts and you may need to scroll down to complete all parts.</p>\n    \n  ", "preamble": {"css": "", "js": ""}, "parts": [{"customMarkingAlgorithm": "", "prompt": "\n        <p>Is $g(x)$ continuous at all points of $I$?</p>\n        \n        <p>Choose Yes or No.</p>\n        \n      ", "variableReplacements": [], "marks": 0, "showCellAnswerState": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "scripts": {}, "displayType": "radiogroup", "showCorrectAnswer": true, "choices": ["<p>Yes</p>", "<p>No</p>"], "adaptiveMarkingPenalty": 0, "displayColumns": 0, "maxMarks": 0, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "shuffleChoices": false, "distractors": ["", ""], "type": "1_n_2", "unitTests": [], "minMarks": 0, "customName": "", "matrix": [1, 0]}, {"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "\n      <p>The first derivative of $g$ can be written in the form $\\displaystyle \\frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)=(x^2+\\var{b^2})^2$ are polynomials.</p>\n        <p>Input the numerator $p(x)$ of the first derivative of $g$ here, factorised into a product of two linear factors in the form <br/>\\[p(x)=c(x-a)(x-b)\\]for suitable integers $a$, $b$ and $c$:</p>\n        <p>$p(x)=\\;\\;$[[0]]</p>\n      ", "variableReplacements": [], "marks": 0, "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "useCustomName": false, "scripts": {}, "type": "gapfill", "gaps": [{"customMarkingAlgorithm": "", "valuegenerators": [{"value": "", "name": "x"}], "variableReplacements": [], "marks": 1, "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "showPreview": true, "useCustomName": false, "scripts": {}, "showCorrectAnswer": true, "checkingAccuracy": 0.001, "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "notallowed": {"showStrings": false, "partialCredit": 0, "strings": ["^", "x*x", "xx", "x x"], "message": "<p>Factorise the expression</p>"}, "musthave": {"showStrings": false, "partialCredit": 0, "strings": ["(", ")"], "message": "<p>Factorise the expression</p>"}, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "showFeedbackIcon": true, "type": "jme", "failureRate": 1, "unitTests": [], "answerSimplification": "std", "customName": "", "answer": "({( - a)} * (x + ( - {b})) * (x + {b}))"}], "showCorrectAnswer": true, "unitTests": [], "adaptiveMarkingPenalty": 0, "customName": ""}, {"customMarkingAlgorithm": "", "prompt": "\n        <p>Is $g(x)$ differentiable at all points of $I$?</p>\n        \n        <p>Choose Yes or No.</p>\n        \n      ", "variableReplacements": [], "marks": 0, "showCellAnswerState": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "scripts": {}, "displayType": "radiogroup", "showCorrectAnswer": true, "choices": ["<p>Yes</p>", "<p>No</p>"], "adaptiveMarkingPenalty": 0, "displayColumns": 0, "maxMarks": 0, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "shuffleChoices": true, "distractors": ["", ""], "type": "1_n_2", "unitTests": [], "minMarks": 0, "customName": "", "matrix": [1, 0]}, {"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "\n        <p>Assume now that $g$ is a function $g:\\mathbb{R} \\rightarrow \\mathbb{R}$.</p>\n        \n        <h4>Find the stationary points of $g$.</h4>\n        \n        <p>Least stationary point: [[0]]</p>\n        \n        <p>Greatest stationary point: [[1]]</p>\n        \n        <p>Are both stationary points in the interval $I$? Choose Yes or No.<br/>[[2]]</p>\n        \n      ", "variableReplacements": [], "marks": 0, "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "useCustomName": false, "scripts": {}, "type": "gapfill", "gaps": [{"customMarkingAlgorithm": "", "valuegenerators": [], "variableReplacements": [], "marks": 1, "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "showPreview": true, "useCustomName": false, "scripts": {}, "showCorrectAnswer": true, "checkingAccuracy": 0.001, "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "showFeedbackIcon": true, "type": "jme", "failureRate": 1, "unitTests": [], "answerSimplification": "std", "customName": "", "answer": "{-b}"}, {"customMarkingAlgorithm": "", "valuegenerators": [], "variableReplacements": [], "marks": 1, "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "showPreview": true, "useCustomName": false, "scripts": {}, "showCorrectAnswer": true, "checkingAccuracy": 0.001, "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "showFeedbackIcon": true, "type": "jme", "failureRate": 1, "unitTests": [], "answerSimplification": "std", "customName": "", "answer": "{b}"}, {"customMarkingAlgorithm": "", "variableReplacements": [], "marks": 0, "showCellAnswerState": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "scripts": {}, "displayType": "radiogroup", "showCorrectAnswer": true, "choices": ["<p>Yes</p>", "<p>No</p>"], "adaptiveMarkingPenalty": 0, "displayColumns": 0, "maxMarks": 0, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "shuffleChoices": true, "distractors": ["", ""], "type": "1_n_2", "unitTests": [], "minMarks": 0, "customName": "", "matrix": [1, 0]}], "showCorrectAnswer": true, "unitTests": [], "adaptiveMarkingPenalty": 0, "customName": ""}, {"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "\n      <p>The second derivative of $g$ can be written in the form $\\displaystyle \\frac{r(x)}{s(x)}$ where $r(x)$ and $s(x)=(x^2+\\var{b^2})^3$ are polynomials.</p>\n        <p>Input the numerator $r(x)$ of the second derivative of $g$ here, factorised into a product of a linear factor and a quadratic factor in the form <br/>\\[r(x)=a_1x(x^2-a_2)\\] for suitable integers $a_1$, $a_2$</p>\n        <p>$r(x)=\\;\\;$ [[0]]</p>\n        <p>Hence find all local maxima and minima given by the stationary points</p>\n        <p>Local maximum is at $x=\\;\\;$ [[1]] and the value of the function at the local maximum (to 3 decimal places)= [[2]]</p>\n        <p>Local minimum is at $x=\\;\\;$ [[3]] and the value of the function at the local minimum (to 3 decimal places) = [[4]]</p>\n      ", "variableReplacements": [], "marks": 0, "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "useCustomName": false, "scripts": {}, "type": "gapfill", "gaps": [{"customMarkingAlgorithm": "", "valuegenerators": [{"value": "", "name": "x"}], "variableReplacements": [], "marks": 1, "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "showPreview": true, "useCustomName": false, "scripts": {}, "showCorrectAnswer": true, "checkingAccuracy": 0.001, "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "notallowed": {"showStrings": false, "partialCredit": 0, "strings": ["x^2", "x x", "xx"], "message": "<p>Factorise the expression</p>"}, "musthave": {"showStrings": false, "partialCredit": 0, "strings": ["(", ")"], "message": "<p>Factorise the expression</p>"}, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "showFeedbackIcon": true, "type": "jme", "failureRate": 1, "unitTests": [], "answerSimplification": "std", "customName": "", "answer": "{2*a}*x*(x^2-{3*b^2})"}, {"customMarkingAlgorithm": "", "valuegenerators": [], "variableReplacements": [], "marks": 1, "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "showPreview": true, "useCustomName": false, "scripts": {}, "showCorrectAnswer": true, "checkingAccuracy": 0.001, "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "showFeedbackIcon": true, "type": "jme", "failureRate": 1, "unitTests": [], "answerSimplification": "std", "customName": "", "answer": "{lma}"}, {"customMarkingAlgorithm": "", "variableReplacements": [], "marks": 1, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "scripts": {}, "showCorrectAnswer": true, "allowFractions": false, "minValue": "{valmax-tol}", "adaptiveMarkingPenalty": 0, "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "showFractionHint": true, "showFeedbackIcon": true, "mustBeReduced": false, "maxValue": "{valmax+tol}", "type": "numberentry", "correctAnswerStyle": "plain", "unitTests": [], "customName": "", "mustBeReducedPC": 0}, {"customMarkingAlgorithm": "", "valuegenerators": [], "variableReplacements": [], "marks": 1, "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "showPreview": true, "useCustomName": false, "scripts": {}, "showCorrectAnswer": true, "checkingAccuracy": 0.001, "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "showFeedbackIcon": true, "type": "jme", "failureRate": 1, "unitTests": [], "answerSimplification": "std", "customName": "", "answer": "{lmi}"}, {"customMarkingAlgorithm": "", "variableReplacements": [], "marks": 1, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "scripts": {}, "showCorrectAnswer": true, "allowFractions": false, "minValue": "{valmin-tol}", "adaptiveMarkingPenalty": 0, "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "showFractionHint": true, "showFeedbackIcon": true, "mustBeReduced": false, "maxValue": "{valmin+tol}", "type": "numberentry", "correctAnswerStyle": "plain", "unitTests": [], "customName": "", "mustBeReducedPC": 0}], "showCorrectAnswer": true, "unitTests": [], "adaptiveMarkingPenalty": 0, "customName": ""}, {"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "\n        <p>What are the following values at the end points of the interval $I$ ?</p>\n        \n        <p>$g(\\var{c})=\\;\\;$ [[0]]</p>\n        \n        <p>$g(\\var{d})=\\;\\;$ [[1]]</p>\n        \n        <p>Input both to 3 decimal places.</p>\n        \n      ", "variableReplacements": [], "marks": 0, "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "useCustomName": false, "scripts": {}, "type": "gapfill", "gaps": [{"customMarkingAlgorithm": "", "variableReplacements": [], "marks": 1, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "scripts": {}, "showCorrectAnswer": true, "allowFractions": false, "minValue": "precround(valbegin-0.001,3)", "adaptiveMarkingPenalty": 0, "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "showFractionHint": true, "showFeedbackIcon": true, "mustBeReduced": false, "maxValue": "precround(valbegin+0.001,3)", "type": "numberentry", "correctAnswerStyle": "plain", "unitTests": [], "customName": "", "mustBeReducedPC": 0}, {"customMarkingAlgorithm": "", "variableReplacements": [], "marks": 1, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "scripts": {}, "showCorrectAnswer": true, "allowFractions": false, "minValue": "precround(valend-0.001,3)", "adaptiveMarkingPenalty": 0, "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "showFractionHint": true, "showFeedbackIcon": true, "mustBeReduced": false, "maxValue": "precround(valend+0.001,3)", "type": "numberentry", "correctAnswerStyle": "plain", "unitTests": [], "customName": "", "mustBeReducedPC": 0}], "showCorrectAnswer": true, "unitTests": [], "adaptiveMarkingPenalty": 0, "customName": ""}, {"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "\n        <p>Assume now that $g: \\mathbb{R} \\rightarrow \\mathbb{R}$ and you are given that:</p>\n        \n        <p>$\\lim_{x \\to \\infty}g(x)=0$ and $\\lim_{x \\to -\\infty}g(x)=0$</p>\n        \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        <h4>Global Minimum</h4>\n        \n        <p>At what value of $x \\in I$ does $g$ have a global minimum ?</p>\n        \n        <p>$x=\\;\\;$ [[1]]</p>\n        \n      ", "variableReplacements": [], "marks": 0, "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "useCustomName": false, "scripts": {}, "type": "gapfill", "gaps": [{"customMarkingAlgorithm": "", "variableReplacements": [], "marks": 1, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "scripts": {}, "showCorrectAnswer": true, "allowFractions": false, "minValue": "{lma}", "adaptiveMarkingPenalty": 0, "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "showFractionHint": true, "showFeedbackIcon": true, "mustBeReduced": false, "maxValue": "{lma}", "type": "numberentry", "correctAnswerStyle": "plain", "unitTests": [], "customName": "", "mustBeReducedPC": 0}, {"customMarkingAlgorithm": "", "variableReplacements": [], "marks": 1, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "scripts": {}, "showCorrectAnswer": true, "allowFractions": false, "minValue": "{lmi}", "adaptiveMarkingPenalty": 0, "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "showFractionHint": true, "showFeedbackIcon": true, "mustBeReduced": false, "maxValue": "{lmi}", "type": "numberentry", "correctAnswerStyle": "plain", "unitTests": [], "customName": "", "mustBeReducedPC": 0}], "showCorrectAnswer": true, "unitTests": [], "adaptiveMarkingPenalty": 0, "customName": ""}], "name": "Paul 's copy of Max and Min 5", "variable_groups": [], "ungrouped_variables": ["a", "c", "b", "d", "valmax", "s", "lmi", "valend", "tol", "valbegin", "lma", "valmin"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "extensions": [], "contributors": [{"name": "Paul Howes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/632/"}]}]}], "contributors": [{"name": "Paul Howes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/632/"}]}