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Let $I=[\\var{c},\\var{d}]$ be an interval and let $g: I \\rightarrow I$ be the function given by:
\\[g(x)=\\simplify{{a}*x/(x^2+{b}^2)}\\]
Answer the following questions. There are seven parts and you may need to scroll down to complete all parts.
\n \n ", "name": "Paul 's copy of Max and Min 5", "parts": [{"customMarkingAlgorithm": "", "matrix": [1, 0], "extendBaseMarkingAlgorithm": true, "customName": "", "variableReplacementStrategy": "originalfirst", "choices": ["Yes
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\n \nChoose Yes or No.
\n \n ", "displayColumns": 0, "marks": 0, "useCustomName": false, "shuffleChoices": false, "maxMarks": 0, "displayType": "radiogroup", "showCorrectAnswer": true, "scripts": {}, "type": "1_n_2", "showFeedbackIcon": true, "unitTests": [], "showCellAnswerState": true, "variableReplacements": []}, {"marks": 0, "customMarkingAlgorithm": "", "scripts": {}, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "\nThe 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.
\nInput the numerator $p(x)$ of the first derivative of $g$ here, factorised into a product of two linear factors in the form
\\[p(x)=c(x-a)(x-b)\\]for suitable integers $a$, $b$ and $c$:
$p(x)=\\;\\;$[[0]]
\n ", "gaps": [{"vsetRangePoints": 5, "customMarkingAlgorithm": "", "showPreview": true, "extendBaseMarkingAlgorithm": true, "customName": "", "musthave": {"showStrings": false, "message": "Factorise the expression
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", "strings": ["^", "x*x", "xx", "x x"], "partialCredit": 0}, "checkingAccuracy": 0.001, "valuegenerators": [{"name": "x", "value": ""}], "showFeedbackIcon": true, "marks": 1, "checkVariableNames": false, "answerSimplification": "std", "vsetRange": [0, 1], "failureRate": 1, "answer": "({( - a)} * (x + ( - {b})) * (x + {b}))", "showCorrectAnswer": true, "scripts": {}, "type": "jme", "unitTests": [], "useCustomName": false, "variableReplacements": []}], "unitTests": [], "useCustomName": false, "variableReplacements": []}, {"customMarkingAlgorithm": "", "matrix": [1, 0], "extendBaseMarkingAlgorithm": true, "customName": "", "variableReplacementStrategy": "originalfirst", "choices": ["Yes
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"], "distractors": ["", ""], "minMarks": 0, "prompt": "\nIs $g(x)$ differentiable at all points of $I$?
\n \nChoose Yes or No.
\n \n ", "displayColumns": 0, "marks": 0, "useCustomName": false, "shuffleChoices": true, "maxMarks": 0, "displayType": "radiogroup", "showCorrectAnswer": true, "scripts": {}, "type": "1_n_2", "showFeedbackIcon": true, "unitTests": [], "showCellAnswerState": true, "variableReplacements": []}, {"marks": 0, "customMarkingAlgorithm": "", "scripts": {}, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "\nAssume now that $g$ is a function $g:\\mathbb{R} \\rightarrow \\mathbb{R}$.
\n \nLeast stationary point: [[0]]
\n \nGreatest stationary point: [[1]]
\n \nAre both stationary points in the interval $I$? Choose Yes or No.
[[2]]
Yes
", "No
"], "minMarks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "1_n_2", "distractors": ["", ""], "displayColumns": 0, "unitTests": [], "useCustomName": false, "shuffleChoices": true, "variableReplacements": []}], "unitTests": [], "useCustomName": false, "variableReplacements": []}, {"marks": 0, "customMarkingAlgorithm": "", "scripts": {}, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "\nThe 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.
\nInput 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
\\[r(x)=a_1x(x^2-a_2)\\] for suitable integers $a_1$, $a_2$
$r(x)=\\;\\;$ [[0]]
\nHence find all local maxima and minima given by the stationary points
\nLocal maximum is at $x=\\;\\;$ [[1]] and the value of the function at the local maximum (to 3 decimal places)= [[2]]
\nLocal minimum is at $x=\\;\\;$ [[3]] and the value of the function at the local minimum (to 3 decimal places) = [[4]]
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\n \n$g(\\var{c})=\\;\\;$ [[0]]
\n \n$g(\\var{d})=\\;\\;$ [[1]]
\n \nInput both to 3 decimal places.
\n \n ", "gaps": [{"correctAnswerStyle": "plain", "marks": 1, "mustBeReduced": false, "customMarkingAlgorithm": "", "scripts": {}, "extendBaseMarkingAlgorithm": true, "customName": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "minValue": "precround(valbegin-0.001,3)", "mustBeReducedPC": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "numberentry", "maxValue": "precround(valbegin+0.001,3)", "allowFractions": false, "unitTests": [], "useCustomName": false, "correctAnswerFraction": false, "variableReplacements": []}, {"correctAnswerStyle": "plain", "marks": 1, "mustBeReduced": false, "customMarkingAlgorithm": "", "scripts": {}, "extendBaseMarkingAlgorithm": true, "customName": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "minValue": "precround(valend-0.001,3)", "mustBeReducedPC": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "numberentry", "maxValue": "precround(valend+0.001,3)", "allowFractions": false, "unitTests": [], "useCustomName": false, "correctAnswerFraction": false, "variableReplacements": []}], "unitTests": [], "useCustomName": false, "variableReplacements": []}, {"marks": 0, "customMarkingAlgorithm": "", "scripts": {}, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "\nAssume now that $g: \\mathbb{R} \\rightarrow \\mathbb{R}$ and you are given that:
\n \n$\\lim_{x \\to \\infty}g(x)=0$ and $\\lim_{x \\to -\\infty}g(x)=0$
\n \nAt what value of $x \\in I$ does $g$ have a global maximum ?
\n \n$x=\\;\\;$ [[0]]
\n \nAt what value of $x \\in I$ does $g$ have a global minimum ?
\n \n$x=\\;\\;$ [[1]]
\n \n ", "gaps": [{"correctAnswerStyle": "plain", "marks": 1, "mustBeReduced": false, "customMarkingAlgorithm": "", "scripts": {}, "extendBaseMarkingAlgorithm": true, "customName": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "minValue": "{lma}", "mustBeReducedPC": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "numberentry", "maxValue": "{lma}", "allowFractions": false, "unitTests": [], "useCustomName": false, "correctAnswerFraction": false, "variableReplacements": []}, {"correctAnswerStyle": "plain", "marks": 1, "mustBeReduced": false, "customMarkingAlgorithm": "", "scripts": {}, "extendBaseMarkingAlgorithm": true, "customName": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "minValue": "{lmi}", "mustBeReducedPC": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "numberentry", "maxValue": "{lmi}", "allowFractions": false, "unitTests": [], "useCustomName": false, "correctAnswerFraction": false, "variableReplacements": []}], "unitTests": [], "useCustomName": false, "variableReplacements": []}], "functions": {}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "tags": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "extensions": [], "ungrouped_variables": ["a", "c", "b", "d", "valmax", "s", "lmi", "valend", "tol", "valbegin", "lma", "valmin"], "advice": "The function $g(x)$ is continuous and differentiable at all points in $\\mathbb{R}$.
\nUsing the quotient rule for differentiation we see that
\\[\\begin{eqnarray*}g'(x)&=&\\simplify{({a}*(x^2+{b^2})-{2*a}*x^2)/(x^2+{b^2})^2}\\\\ &=&\\simplify{({-a}*(x-{b})(x+{b}))/(x^2+{b^2})^2} \\end{eqnarray*} \\]
The stationary points are given by solving $g'(x)=0$.
\n$g'(x)=0 \\Rightarrow \\simplify{{-a}*(x-{b})(x+{b})=0} \\Rightarrow x=\\var{b} \\mbox{ or } x=\\var{-b}$
\nWe see that both stationary points are in the inerval $I$.
\nThe second derivative can be found by applying the quotient rule to the derivative of $g(x)$ and we obtain:
\nUsing the quotient rule for differentiation we see that
\\[\\begin{eqnarray*}g''(x)&=&\\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}\\\\ &=&\\simplify[std]{({2*a}*x*(x^2-{3*b^2}))/(x^2+{b^2})^3} \\end{eqnarray*} \\]
The nature of the stationary points are determined by evaluating $g''(x)$ at the stationary points.
\nFor $x= \\var{lma}$ we have: \\[g''(\\var{lma})= \\simplify[std]{-{abs(a)}/{2*b^3}} \\lt 0\\]
\nHence is a local maximum.
\nEvaluating the function at $x=\\var{lma}$ gives $g(\\var{lma})=\\var{valmax}$ to 3 decimal places.
\nFor $x= \\var{lmi}$ we have: \\[g''(\\var{lmi})= \\simplify[std]{{abs(a)}/{2*b^3}} \\gt 0\\]
\nHence is a local minimum.
\nEvaluating the function at $x=\\var{lmi}$ gives $g(\\var{lmi})=\\var{valmin}$ to 3 decimal places.
\nThe values of $g$ at the endpoints are:
\n$g(\\var{c})=\\var{valbegin}$ and $g(\\var{d})=\\var{valend}$ to 3 decimal places.
\nSince $g$ 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 $I$).
\nthen:
\nGlobal Maximum: The local maximum of $g$ we have found at $x=\\var{lma} \\in I$ must be a global maximum and similarly,
\nGlobal Minimum: The local minimum of $g$ we have found at $x=\\var{lmi} \\in I$ must be a global minimum.
\nSo we have shown \\[\\forall x \\in \\mathbb{R},\\;\\;\\var{valmin} \\le g(x) \\le \\var{valmax}\\]
\n(all to 3 decimal places).
", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "$I$ compact interval. $\\displaystyle g: I \\rightarrow I, g(x)=\\frac{ax}{x^2+b^2}$. Find stationary points and local maxima, minima. Using limits, has $g$ a global max, min?
"}, "contributors": [{"name": "Paul Howes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/632/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Paul Howes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/632/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}