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{"name": "Paul 's copy of Max and Min 4", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "b", "valmin", "valmax", "lmi", "s", "tol", "lma"], "name": "Paul 's copy of Max and Min 4", "tags": ["Calculus", "calculus", "classifying stationary points", "functions", "global maxima and minima", "limit", "limits", "local maxima and minima", "maxima and minima", "optimisation", "optimising", "optimising functions", "rational polynomials", "stationary points", "taking limits"], "preamble": {"css": "", "js": ""}, "advice": "\n  <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>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    <h4>The Limits.</h4>\n    <p>If we divide $g(x)$ top and bottom by $x^2$ (OK as $x \\neq 0$ at any time) we obtain: \\[g(x)=\\simplify[std]{({a}/x)/(1+{b^2}/x^2)}\\]</p>\n    <p>Then using the fact that $\\displaystyle \\frac{1}{x}$ and $\\displaystyle \\frac{1}{x^2}$ both tend to $0$ as $ x \\rightarrow \\pm\\infty$ we see that</p>\n    <p>$\\displaystyle \\lim_{x \\to \\infty}g(x)=\\frac{0}{1}=0$ and similarly</p>\n    <p>$\\lim_{x \\to -\\infty}g(x)=0$</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 $\\var{valmin}$ and the local maximum $\\var{valmax}$</p>\n    <p>Then:</p>\n    <p><strong>Global Maximum</strong>: The local maximum of $g$ we have found at $x=\\var{lma}$ must be a global maximum and similarly,</p>\n    <p><strong>Global Minimum</strong>: The local minimum of $g$ we have found at $x=\\var{lmi}$ 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  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n      <p>Is $g(x)$ continuous at all points of $\\mathbb{R}$?</p>\n      \n      <p>Choose Yes or No.</p>\n      \n      ", "matrix": [1, 0], "shuffleChoices": false, "scripts": {}, "choices": ["<p>Yes</p>", "<p>No</p>"], "marks": 0, "displayType": "radiogroup", "maxMarks": 0, "distractors": ["", ""], "displayColumns": 0, "showCorrectAnswer": true, "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"}, {"prompt": "\n      <p>Is $g(x)$ differentiable at all points of $\\mathbb{R}$?</p>\n      \n      <p>Choose Yes or No.</p>\n      \n      ", "matrix": [1, 0], "shuffleChoices": true, "scripts": {}, "choices": ["<p>Yes</p>", "<p>No</p>"], "marks": 0, "displayType": "radiogroup", "maxMarks": 0, "distractors": ["", ""], "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}, {"prompt": "\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      ", "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"}], "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 as asked in the question.</p>", "showStrings": false, "strings": ["x^3"], "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 limits?</p>\n      <p>1) $\\lim_{x \\to \\infty}g(x)\\;\\;$</p>\n      <p>Choose one of the following [[0]]</p>\n      <p>2) $\\lim_{x \\to -\\infty}g(x)$</p>\n      <p>Choose one of the following [[1]]</p>\n      <p>Does $g$ have a finite global maximum? Click on Yes or No <br/>[[2]]</p>\n      <p>Does $g$ have a finite global maximum? Click on Yes or No <br/>[[2]]</p>\n      ", "marks": 0, "gaps": [{"matrix": [0, 0, 0, 0, 1], "shuffleChoices": true, "scripts": {}, "choices": ["<p>$-\\infty$</p>", "<p>$\\infty$</p>", "<p>$\\var{b}$</p>", "<p>$\\var{valmax}$</p>", "<p>$0$</p>"], "marks": 0, "displayType": "radiogroup", "maxMarks": 0, "distractors": ["", "", "", "", ""], "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}, {"matrix": [0, 0, 0, 0, 1], "shuffleChoices": true, "scripts": {}, "choices": ["<p>$-\\infty$</p>", "<p>$\\infty$</p>", "<p>$\\var{a}$</p>", "<p>$\\var{valmin}$</p>", "<p>$0$</p>"], "marks": 0, "displayType": "radiogroup", "maxMarks": 0, "distractors": ["", "", "", "", ""], "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}, {"matrix": [1, 0], "shuffleChoices": true, "scripts": {}, "choices": ["<p>Yes</p>", "<p>No</p>"], "marks": 0, "displayType": "radiogroup", "maxMarks": 0, "distractors": ["", ""], "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}, {"matrix": [1, 0], "shuffleChoices": true, "scripts": {}, "choices": ["<p>Yes</p>", "<p>No</p>"], "marks": 0, "displayType": "radiogroup", "maxMarks": 0, "distractors": ["", ""], "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "<p>Let $g: \\mathbb{R} \\rightarrow \\mathbb{R}$ be the function given by:<br/>\\[g(x)=\\simplify{{a}*x/(x^2+{b}^2)}\\]</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "type": "question", "variables": {"a": {"definition": "s*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "valmin": {"definition": "precround(-abs(a)*b/(2*b^2),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "valmin", "description": ""}, "valmax": {"definition": "-valmin", "templateType": "anything", "group": "Ungrouped variables", "name": "valmax", "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": ""}, "lma": {"definition": "if(a>0,b,-b)", "templateType": "anything", "group": "Ungrouped variables", "name": "lma", "description": ""}}, "metadata": {"notes": "\n    \t\t<p><strong>10/07/2012:<br /></strong></p>\n    \t\t<p>Added tags.<strong><br /></strong></p>\n    \t\t<p>Question appears to be working correctly.</p>\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", "description": "<p>$g: \\mathbb{R} \\rightarrow \\mathbb{R},&nbsp;g(x)=\\frac{ax}{x^2+b^2}$. 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": "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/"}]}