// Numbas version: exam_results_page_options
{"name": "Max and Min (5 parts) 24/11", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "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"], "name": "Max and Min (5 parts) 24/11", "tags": ["Calculus", "calculus", "classifying stationary points", "finding global maxima and minima", "finding local maxima and local minima", "finding stationary points", "finding the maximum and minimum of a function", "nature of a critical point", "nature of critical points", "nature of turning points", "optimisation", "optimising a function on an interval", "optimising functions", "stationary points", "turning points"], "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>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>In our example we see that the global minimum occurs at $x=\\var{gmi}$ and $g(\\var{gmi})=\\var{valgmin}$.</p>\n<p></p>\n<p>More information on maxima and minima is chapter 3 of your Mathematical Physics book, sections 3.2 and 3.3, pages 51-53.</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "<p>Input the first derivative of $g$ here, factorised into a product of two linear factors in the form $g'(x)=c(x-a)(x-b)$for suitable integers $a$, $b$ and $c$ (these can be zero):</p>\n<p>$g'(x)=\\;\\;$[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>Factorise the expression</p>", "showStrings": false, "strings": ["^", "x*x", "xx", "x x"], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{c} * (x + {-a}) * (x + {-b})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1], "musthave": {"message": "<p>Factorise the expression</p>", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<h4>Find the stationary points of $g$.</h4>\n<p>Least stationary point is at $x$ = [[0]] &nbsp; &nbsp;Greatest stationary point is at $x$ = [[1]]</p>\n<p>Do both these stationary points lie in the interval $I$ ? [[2]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{a}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{b}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"matrix": [1, 0], "shuffleChoices": true, "marks": 0, "variableReplacements": [], "choices": ["<p>Yes</p>", "<p>No</p>"], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "maxMarks": 0, "scripts": {}, "distractors": ["", ""], "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"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      ", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "(({(2 * c)} * x) + ( - {(c * (a + b))}))", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{a}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{valmax}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{b}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{valmin}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "\n      <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 3 decimal places.</p>\n      ", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "valbegin", "minValue": "valbegin", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "valend", "minValue": "valend", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<h4>Global Maximum</h4>\n<p>At what value of $x$ does $g$ have a global maximum ?</p>\n<p>$x=\\;\\;$ [[0]]</p>\n<p>Value of $g$ at this global maximum = [[1]] (input to 3 decimal places).</p>\n<h4>Global Minimum</h4>\n<p>At what value of $x$ does $g$ have a global minimum ?</p>\n<p>$x=\\;\\;$ [[2]]</p>\n<p>Value of $g$ at this global minimum = [[3]] (input to 3 decimal places).</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{gma}", "minValue": "{gma}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "valgmax", "minValue": "valgmax", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{gmi}", "minValue": "{gmi}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "valgmin", "minValue": "valgmin", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>Let $I=[\\var{l},\\var{m}]$ be an interval in $x$ and let $g$ be a function defined on this interval given by :\\[g(x) = \\simplify{{c}/3*x^3+ {-c*(a+b)}/2*x^2+{c*a*b}*x+{d}}\\]</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"temp2": {"definition": "precround(rtemp2,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "temp2", "description": ""}, "temp1": {"definition": "precround(rtemp1,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "temp1", "description": ""}, "gmi": {"definition": "if(valbegin<valmin,l,b)", "templateType": "anything", "group": "Ungrouped variables", "name": "gmi", "description": ""}, "valmin": {"definition": "round(rawvalmin)", "templateType": "anything", "group": "Ungrouped variables", "name": "valmin", "description": ""}, "gma": {"definition": "if(valend>valmax,m,a)", "templateType": "anything", "group": "Ungrouped variables", "name": "gma", "description": ""}, "valend": {"definition": "precround(rawvalend,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "valend", "description": ""}, "m1": {"definition": "b+random(0.5..3#0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "m1", "description": ""}, "valbegin": {"definition": "precround(rawvalbegin,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "valbegin", "description": ""}, "rtemp2": {"definition": "(c/3) * m1^ 3 + (-c * (a + b)) / 2 * m1^ 2 + (c * a * b) * m1+ d", "templateType": "anything", "group": "Ungrouped variables", "name": "rtemp2", "description": ""}, "valmax": {"definition": "round(rawvalmax)", "templateType": "anything", "group": "Ungrouped variables", "name": "valmax", "description": ""}, "valgmax": {"definition": "max(valend,valmax)", "templateType": "anything", "group": "Ungrouped variables", "name": "valgmax", "description": ""}, "valgmin": {"definition": "min(valbegin,valmin)", "templateType": "anything", "group": "Ungrouped variables", "name": "valgmin", "description": ""}, "rawvalend": {"definition": "(c/3) * m^ 3 + (-c * (a + b)) / 2 * m^ 2 + (c * a * b) * m+ d", "templateType": "anything", "group": "Ungrouped variables", "name": "rawvalend", "description": ""}, "rawvalmax": {"definition": "(c/3) * a^ 3 + (-c * (a + b)) / 2 * a^ 2 + (c * a * b) * a + d", "templateType": "anything", "group": "Ungrouped variables", "name": "rawvalmax", "description": ""}, "b": {"definition": "if(c=6,a+random(1..9),random(2..10#2)+a)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "rawvalbegin": {"definition": "(c/3) * l^ 3 + (-c * (a + b)) / 2 * l^ 2 + (c * a * b) * l+ d", "templateType": "anything", "group": "Ungrouped variables", "name": "rawvalbegin", "description": ""}, "rawvalmin": {"definition": "(c/3) * b^ 3 + (-c * (a + b)) / 2 * b^ 2 + (c * a * b) * b + d", "templateType": "anything", "group": "Ungrouped variables", "name": "rawvalmin", "description": ""}, "a": {"definition": "s*random(1..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(3,6)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "rtemp1": {"definition": "(c/3) * l1^ 3 + (-c * (a + b)) / 2 * l1^ 2 + (c * a * b) * l1+ d", "templateType": "anything", "group": "Ungrouped variables", "name": "rtemp1", "description": ""}, "d": {"definition": "if(c=3,random(-9..9),random(-9..9#2))", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "m": {"definition": "if(valmax=temp2,m1+0.5,m1)", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "l": {"definition": "if(valmin=temp1,l1-0.5,l1)", "templateType": "anything", "group": "Ungrouped variables", "name": "l", "description": ""}, "s": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "l1": {"definition": "a-random(0.5..3#0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "l1", "description": ""}}, "metadata": {"notes": "\n    \t\t<p><strong>9/07/2102:</strong></p>\n    \t\t<p>Added tags.</p>\n    \t\t<p>Question appears to be working correctly.</p>\n    \t\t<p>Changed grammar in the Advice section.</p>\n    \t\t", "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"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Katie Dexter", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/553/"}]}]}], "contributors": [{"name": "Katie Dexter", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/553/"}]}