// Numbas version: exam_results_page_options
{"name": "Max and Min 5", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "c", "b", "d", "valmax", "s", "lmi", "valend", "tol", "valbegin", "lma", "valmin"], "name": "Max and Min 5", "tags": ["Calculus", "calculus", "classifying stationary points", "finding local maxima and minima", "finding the stationary points", "limit", "limits", "maxima and minima", "maximum", "min", "minimum", "optimisation", "optimising functions on an interval", "stationary points", "taking limits"], "preamble": {"css": "", "js": ""}, "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>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"displayColumns": 0, "prompt": "\n        <p>Is $g(x)$ continuous at all points of $I$?</p>\n        \n        <p>Choose Yes or No.</p>\n        \n      ", "matrix": [1, 0], "shuffleChoices": false, "maxMarks": 0, "distractors": ["", ""], "choices": ["<p>Yes</p>", "<p>No</p>"], "displayType": "radiogroup", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "1_n_2", "minMarks": 0}, {"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      ", "marks": 0, "gaps": [{"notallowed": {"message": "<p>Factorise the expression</p>", "showStrings": false, "strings": ["^", "x*x", "xx", "x x"], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "({( - a)} * (x + ( - {b})) * (x + {b}))", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme", "musthave": {"message": "<p>Factorise the expression</p>", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"displayColumns": 0, "prompt": "\n        <p>Is $g(x)$ differentiable at all points of $I$?</p>\n        \n        <p>Choose Yes or No.</p>\n        \n      ", "matrix": [1, 0], "shuffleChoices": true, "maxMarks": 0, "distractors": ["", ""], "choices": ["<p>Yes</p>", "<p>No</p>"], "displayType": "radiogroup", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "1_n_2", "minMarks": 0}, {"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      ", "marks": 0, "gaps": [{"expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{-b}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{b}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"displayColumns": 0, "matrix": [1, 0], "shuffleChoices": true, "maxMarks": 0, "distractors": ["", ""], "choices": ["<p>Yes</p>", "<p>No</p>"], "displayType": "radiogroup", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "1_n_2", "minMarks": 0}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"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      ", "marks": 0, "gaps": [{"notallowed": {"message": "<p>Factorise the expression</p>", "showStrings": false, "strings": ["x^2", "x x", "xx"], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{2*a}*x*(x^2-{3*b^2})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme", "musthave": {"message": "<p>Factorise the expression</p>", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}}, {"expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{lma}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"allowFractions": false, "marks": 1, "maxValue": "{valmax+tol}", "minValue": "{valmax-tol}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{lmi}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"allowFractions": false, "marks": 1, "maxValue": "{valmin+tol}", "minValue": "{valmin-tol}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"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      ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "precround(valbegin+0.001,3)", "minValue": "precround(valbegin-0.001,3)", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "precround(valend+0.001,3)", "minValue": "precround(valend-0.001,3)", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"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      ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "{lma}", "minValue": "{lma}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "{lmi}", "minValue": "{lmi}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "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  ", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "s*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "-b-random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "b+random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "valmax": {"definition": "-valmin", "templateType": "anything", "group": "Ungrouped variables", "name": "valmax", "description": ""}, "valend": {"definition": "precround(a*d/(d^2+b^2),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "valend", "description": ""}, "lmi": {"definition": "if(a<0,b,-b)", "templateType": "anything", "group": "Ungrouped variables", "name": "lmi", "description": ""}, "s": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "tol": {"definition": "0.001", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "valbegin": {"definition": "precround(a*c/(c^2+b^2),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "valbegin", "description": ""}, "lma": {"definition": "if(a>0,b,-b)", "templateType": "anything", "group": "Ungrouped variables", "name": "lma", "description": ""}, "valmin": {"definition": "precround(-abs(a)*b/(2*b^2),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "valmin", "description": ""}}, "metadata": {"notes": "\n    \t\t<p><strong>9/07/2012:</strong></p>\n    \t\t<p>Added tags.</p>\n    \t\t<p>Corrected mistake in Advice ($x$ instead of $x^2$).</p>\n    \t\t<p>Tolerance variable set to tol=0.001 for a numeric entry.</p>\n    \t\t<p>&nbsp;</p>\n    \t\t<p><strong>10/07/2012:<br /></strong></p>\n    \t\t<p>Added tags.</p>\n    \t\t<p>Edited grammar in the Advice section.<strong><br /></strong></p>\n    \t\t<p>Question appears to be working correctly.</p>\n    \t\t", "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>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}