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{"name": "Paul 's copy of Max and Min 6", "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", "valmin", "m", "l", "xma", "lmi", "s", "u", "valend", "tol", "l1", "valbegin", "lma", "statpoint", "xmi", "valsd"], "name": "Paul 's copy of Max and Min 6", "tags": ["Calculus", "Differentiation", "calculus", "classifying stationary points", "continuous functions", "derivative", "derivatives", "differentiable functions", "differentiate", "differentiation", "first derivative", "functions", "global maximum", "global minimum", "local maximum", "local minimum", "max and min", "maxima", "maximum", "maximum and minimum", "maximum and minimum of a function on an interval", "minima", "minimum", "optimising a function on an interval", "quadratic", "quadratics", "quotient rule", "second derivative", "solution of a quadratic", "stationary points"], "preamble": {"css": "", "js": ""}, "advice": "\n  <p>The function $g(x)$ is continuous and differentiable at all points in $\\mathbb{R}$ as $ \\var{c} \\notin I$.</p>\n  <p>On differentiating we see that<br/>\\[\\begin{eqnarray*}g'(x)&amp;=&amp;\\simplify{2*x/(x-{c})^({a}/{b}) - {a}*x^2/({b}(x-{c})^({a+b}/{b}))}\\\\ &amp;=&amp;\\simplify{({2*b}x*(x-{c})-{a}*x^2)/({b}(x-{c})^({a+b}/{b}))}\\\\ &amp;=&amp;\\simplify{(x*({2*b-a}x-{2*b*c}))/({b}(x-{c})^({a+b}/{b}))} \\end{eqnarray*} \\]</p>\n  <h4>Stationary Points.</h4>\n  <p>The stationary points are given by solving $g'(x)=0$.</p>\n  <p>$\\displaystyle g'(x)=0 \\Rightarrow \\simplify{x*({2*b-a}x-{2*b*c})=0} \\Rightarrow x=0 \\mbox{ or } x=\\simplify[std]{{2*b*c}/{2*b-a}}$</p>\n  <p>We see that $\\displaystyle x=\\simplify[std]{{2*b*c}/{2*b-a}}$ is the only stationary point in $I$.</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/>\\[g''(x)=\\simplify[std]{({a^2-3*a*b+2*b^2}*x^2+{4*b*c*(a-b)}*x+{2*c^2*b^2})/({b^2}(x-{c})^({a+2*b}/{b}))}\\]</p>\n  <p>Hence \\[r(x)=\\simplify[std]{({a^2-3*a*b+2*b^2}*x^2+{4*b*c*(a-b)}*x+{2*c^2*b^2})}\\]</p>\n  <p>The nature of the stationary points are determined by evaluating $g''(x)$ at the stationary points.</p>\n  <p>There is only one stationary point $\\displaystyle &nbsp;x=\\simplify[std]{{2*b*c}/{2*b-a}}$ in $I$ and at that point we have: <br/>\\[g''\\left(\\simplify[std]{{2*b*c}/{2*b-a}}\\right)=\\var{valsd} \\gt 0\\]</p>\n  <p>Hence this point is a local minimum.</p>\n  <h4>Evaluating at end points of the interval.</h4>\n  <p>The values of $g$ at the endpoints are:</p>\n  <p>$g(\\var{l})=\\var{valbegin}$ and $g(\\var{m})=\\var{valend}$, both to 3 decimal places.</p>\n  <h4>Global Maximum and Minimum</h4>\n  <p><strong>Global Maximum</strong>: Since $g$ does not have a local maximum in the interval $I$, it must take a global maximum value at one of the end points of the interval.</p>\n  <p>From the values found for $g(\\var{l})$ and $g(\\var{m})$ found above, we see that $x=\\var{xma}$ is the global maximum for $g$ in the interval $I$.</p>\n  <p><strong>Global Minimum</strong>: The local minimum of $g$ given by $ \\displaystyle x=\\simplify[std]{{2*b*c}/{2*b-a}} \\in I$ is the only local minimum and must be a global minimum in $I$.</p>\n  <p>Note that the global minimum value for $g$ on $I$ is:</p>\n  <p>\\[g\\left(\\simplify[std]{{2*b*c}/{2*b-a}}\\right)=\\var{valmin}\\]</p>\n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"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, "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 $p(x)$ is a polynomial of degree $2$ and $q(x)=\\simplify{{b}*(x-{c})^({a+b}/{b})}$.</p>\n        <p>Input the numerator $p(x)$ of the first derivative of $g$ here:</p>\n        <p>$p(x)=\\;\\;$[[0]]</p>\n      ", "marks": 0, "gaps": [{"expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "x*({2*b-a}*x-{2*b*c})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n      <p>Is $g(x)$ differentiable at all interior points of $I$?</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      <p>Assume now that $g$ is a function $g:\\mathbb{R} \\backslash \\{\\var{c}\\} \\rightarrow \\mathbb{R}$.</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]] (Input as a fraction or an integer and not as a decimal)</p>\n      \n      <p>Which stationary point is in the interval $I$? Choose one of the following:<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": "{0}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"notallowed": {"message": "<p>Input as a fraction or an integer and not as a decimal</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{2*b*c}/{2*b-a}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"matrix": [1, 0], "shuffleChoices": true, "scripts": {}, "choices": ["<p>The greatest</p>", "<p>The least</p>"], "marks": 0, "displayType": "radiogroup", "maxMarks": 0, "distractors": ["", ""], "displayColumns": 0, "showCorrectAnswer": true, "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 $r(x)$ is a quadratic polynomial and $s(x)=\\simplify{{b^2}(x-{c})^({a+2*b}/{b})}$.</p>\n        <p>Input the numerator $r(x)$ of the second derivative of $g$ here:</p>\n        <p>$r(x)=\\;\\;$ [[0]]</p>\n        <p>Hence determine the type of the stationary point which lies in $I$. Choose one of the following:<br/>[[1]]</p>\n      ", "marks": 0, "gaps": [{"expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{a^2-3*a*b+2*b^2}*x^2+{4*b*c*(a-b)}*x+{2*c^2*b^2}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"matrix": [0, 1], "shuffleChoices": true, "scripts": {}, "choices": ["<p>Local maximum</p>", "<p>Local minimum</p>"], "marks": 0, "displayType": "radiogroup", "maxMarks": 0, "distractors": ["", ""], "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}], "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>$g(\\var{l})=\\;\\;$ [[0]]</p>\n      \n      <p>$g(\\var{m})=\\;\\;$ [[1]]</p>\n      \n      <p>Input both to 3 decimal places.</p>\n      \n      ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "valbegin+tol", "minValue": "valbegin-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "valend+tol", "minValue": "valend-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n      <h4>Global Maximum</h4>\n      \n      <p>At what value of $x \\in I$  does $g$ have a global maximum in $I$?</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 in $I$ ?</p>\n      \n      <p>$x=\\;\\;$ [[1]] (Input as a fraction or an integer and not as a decimal)</p>\n      \n      ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "{xma}", "minValue": "{xma}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"notallowed": {"message": "<p>Input as a fraction or an integer and not as a decimal</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{2*b*c}/{2*b-a}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "\n  <p>Let $I=[\\var{l},\\var{m}]$ be an interval and let $g: I \\rightarrow \\mathbb{R}$ be the function given by:<br/>\\[g(x)=\\simplify{x^2/(x-{c})^({a}/{b})}\\]</p>\n  <p>Answer the following questions. There are seven parts and you may need to scroll down to complete all parts.</p>\n  <p>&nbsp;</p>\n  ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "type": "question", "variables": {"a": {"definition": "if(b=3,random(1,2,4,5),if(b=5,random(1,2,3,4,6,7,8,9),random(2,3,4,5,6,8,9)))", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "round(2b/a)+random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(3,5,7)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "valmin": {"definition": "precround(statpoint^2/(statpoint-c)^(a/b),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "valmin", "description": ""}, "m": {"definition": "round(statpoint)+random(2..6)", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "l": {"definition": "if(l1=c,l1+1,l1)", "templateType": "anything", "group": "Ungrouped variables", "name": "l", "description": ""}, "xma": {"definition": "if(valend>valbegin,m,l)", "templateType": "anything", "group": "Ungrouped variables", "name": "xma", "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": ""}, "valend": {"definition": "precround(m^2/(m-c)^(a/b),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "valend", "description": ""}, "u": {"definition": "random(0..90)", "templateType": "anything", "group": "Ungrouped variables", "name": "u", "description": ""}, "valsd": {"definition": "precround(2*(2*b/a-1)^(a/b+2)/c^(a/b),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "valsd", "description": ""}, "tol": {"definition": "0.001", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "l1": {"definition": "round((u*(c+1)+(100-u)*round(statpoint-1))/100)", "templateType": "anything", "group": "Ungrouped variables", "name": "l1", "description": ""}, "valbegin": {"definition": "precround(l^2/(l-c)^(a/b),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "valbegin", "description": ""}, "lma": {"definition": "if(a>0,b,-b)", "templateType": "anything", "group": "Ungrouped variables", "name": "lma", "description": ""}, "xmi": {"definition": "lmi", "templateType": "anything", "group": "Ungrouped variables", "name": "xmi", "description": ""}, "statpoint": {"definition": "2*b*c/(2*b-a)", "templateType": "anything", "group": "Ungrouped variables", "name": "statpoint", "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 definition of variable valsd. Changed the number of decimal places to 5 for this variable as can be very small and positive.</p>\n    \t\t<p>Modified display in Advice slightly.</p>\n    \t\t<p>Set new variable tolerance to be tol=0.001 for entries to 3 dps.</p>\n    \t\t<p>&nbsp;</p>\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>In Advice section, increased size of brackets so that they were big enough to contain a fraction.</p>\n    \t\t<p>Question appears to be working correctly.<strong><br /></strong></p>\n    \t\t<p>&nbsp;</p>\n    \t\t", "description": "<p>$I$ compact interval. $\\displaystyle g: I\\rightarrow I,&nbsp;g(x)=\\frac{x^2}{(x-c)^{a/b}}$. Are there stationary points and local maxima, minima? Has $g$ a global max, global 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/"}]}