The function $g(x)$ is continuous and differentiable at all points in $\\mathbb{R}$ as $ \\var{c} \\notin I$.
\nOn differentiating we see that
\\[\\begin{eqnarray*}g'(x)&=&\\simplify{2*x/(x-{c})^({a}/{b}) - {a}*x^2/({b}(x-{c})^({a+b}/{b}))}\\\\ &=&\\simplify{({2*b}x*(x-{c})-{a}*x^2)/({b}(x-{c})^({a+b}/{b}))}\\\\ &=&\\simplify{(x*({2*b-a}x-{2*b*c}))/({b}(x-{c})^({a+b}/{b}))} \\end{eqnarray*} \\]
The stationary points are given by solving $g'(x)=0$.
\n$\\displaystyle g'(x)=0 \\Rightarrow \\simplify{x*({2*b-a}x-{2*b*c})=0} \\Rightarrow x=0 \\mbox{ or } x=\\simplify[std]{{2*b*c}/{2*b-a}}$
\nWe see that $\\displaystyle x=\\simplify[std]{{2*b*c}/{2*b-a}}$ is the only stationary point in $I$.
\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
\\[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}))}\\]
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})}\\]
\nThe nature of the stationary points are determined by evaluating $g''(x)$ at the stationary points.
\nThere is only one stationary point $\\displaystyle x=\\simplify[std]{{2*b*c}/{2*b-a}}$ in $I$ and at that point we have:
\\[g''\\left(\\simplify[std]{{2*b*c}/{2*b-a}}\\right)=\\var{valsd} \\gt 0\\]
Hence this point is a local minimum.
\nThe values of $g$ at the endpoints are:
\n$g(\\var{l})=\\var{valbegin}$ and $g(\\var{m})=\\var{valend}$, both to 3 decimal places.
\nGlobal Maximum: 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.
\nFrom 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 $I$.
\nGlobal Minimum: 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 $I$.
\nNote that the global minimum value for $g$ on $I$ is:
\n\\[g\\left(\\simplify[std]{{2*b*c}/{2*b-a}}\\right)=\\var{valmin}\\]
\n ", "variablesTest": {"condition": "", "maxRuns": 100}, "extensions": [], "name": "Paul 's copy of Max and Min 6", "parts": [{"unitTests": [], "useCustomName": false, "marks": 0, "maxMarks": 0, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "displayType": "radiogroup", "minMarks": 0, "distractors": ["", ""], "showCellAnswerState": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "choices": ["Yes
", "No
"], "scripts": {}, "displayColumns": 0, "variableReplacements": [], "customName": "", "prompt": "\nIs $g(x)$ continuous at all points of $I$?
\n \nChoose Yes or No.
\n \n ", "matrix": [1, 0], "shuffleChoices": false, "type": "1_n_2", "variableReplacementStrategy": "originalfirst"}, {"customMarkingAlgorithm": "", "useCustomName": false, "scripts": {}, "marks": 0, "variableReplacements": [], "gaps": [{"checkingType": "absdiff", "useCustomName": false, "answerSimplification": "std", "marks": 1, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "valuegenerators": [{"name": "x", "value": ""}], "showPreview": true, "scripts": {}, "variableReplacements": [], "customName": "", "unitTests": [], "answer": "x*({2*b-a}*x-{2*b*c})", "variableReplacementStrategy": "originalfirst"}], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "prompt": "\nThe 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 $q(x)=\\simplify{{b}*(x-{c})^({a+b}/{b})}$.
\nInput the numerator $p(x)$ of the first derivative of $g$ here:
\n$p(x)=\\;\\;$[[0]]
\n ", "unitTests": [], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true}, {"unitTests": [], "useCustomName": false, "marks": 0, "maxMarks": 0, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "displayType": "radiogroup", "minMarks": 0, "distractors": ["", ""], "showCellAnswerState": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "choices": ["Yes
", "No
"], "scripts": {}, "displayColumns": 0, "variableReplacements": [], "customName": "", "prompt": "\nIs $g(x)$ differentiable at all interior points of $I$?
\n \nChoose Yes or No.
\n \n ", "matrix": [1, 0], "shuffleChoices": true, "type": "1_n_2", "variableReplacementStrategy": "originalfirst"}, {"customMarkingAlgorithm": "", "useCustomName": false, "scripts": {}, "marks": 0, "variableReplacements": [], "gaps": [{"checkingType": "absdiff", "useCustomName": false, "answerSimplification": "std", "marks": 1, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "valuegenerators": [], "showPreview": true, "scripts": {}, "variableReplacements": [], "customName": "", "unitTests": [], "answer": "{0}", "variableReplacementStrategy": "originalfirst"}, {"checkingType": "absdiff", "useCustomName": false, "answerSimplification": "std", "marks": 1, "checkingAccuracy": 0.001, "notallowed": {"message": "Input as a fraction or an integer and not as a decimal
", "strings": ["."], "partialCredit": 0, "showStrings": false}, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "valuegenerators": [], "showPreview": true, "scripts": {}, "variableReplacements": [], "customName": "", "unitTests": [], "answer": "{2*b*c}/{2*b-a}", "variableReplacementStrategy": "originalfirst"}, {"displayColumns": 0, "customMarkingAlgorithm": "", "matrix": [1, 0], "choices": ["The greatest
", "The least
"], "useCustomName": false, "scripts": {}, "marks": 0, "variableReplacements": [], "maxMarks": 0, "displayType": "radiogroup", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "showCellAnswerState": true, "unitTests": [], "shuffleChoices": true, "minMarks": 0, "distractors": ["", ""], "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "showFeedbackIcon": true}], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "prompt": "\nAssume now that $g$ is a function $g:\\mathbb{R} \\backslash \\{\\var{c}\\} \\rightarrow \\mathbb{R}$.
\n \nLeast stationary point: [[0]]
\n \nGreatest stationary point: [[1]] (Input as a fraction or an integer and not as a decimal)
\n \nWhich stationary point is in the interval $I$? Choose one of the following:
[[2]]
Local maximum
", "Local minimum
"], "useCustomName": false, "scripts": {}, "marks": 0, "variableReplacements": [], "maxMarks": 0, "displayType": "radiogroup", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "showCellAnswerState": true, "unitTests": [], "shuffleChoices": true, "minMarks": 0, "distractors": ["", ""], "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "showFeedbackIcon": true}], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "prompt": "\nThe second derivative of $g$ can be written in the form $\\displaystyle \\frac{r(x)}{s(x)}$ where $r(x)$ is a quadratic polynomial and $s(x)=\\simplify{{b^2}(x-{c})^({a+2*b}/{b})}$.
\nInput the numerator $r(x)$ of the second derivative of $g$ here:
\n$r(x)=\\;\\;$ [[0]]
\nHence determine the type of the stationary point which lies in $I$. Choose one of the following:
[[1]]
What are the following values at the end points of the interval $I$ ?
\n \n$g(\\var{l})=\\;\\;$ [[0]]
\n \n$g(\\var{m})=\\;\\;$ [[1]]
\n \nInput both to 3 decimal places.
\n \n ", "unitTests": [], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true}, {"customMarkingAlgorithm": "", "useCustomName": false, "scripts": {}, "marks": 0, "variableReplacements": [], "gaps": [{"correctAnswerFraction": false, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "useCustomName": false, "scripts": {}, "marks": 1, "variableReplacements": [], "allowFractions": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "mustBeReduced": false, "unitTests": [], "maxValue": "{xma}", "showFractionHint": true, "minValue": "{xma}", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true}, {"checkingType": "absdiff", "useCustomName": false, "answerSimplification": "std", "marks": 1, "checkingAccuracy": 0.001, "notallowed": {"message": "Input as a fraction or an integer and not as a decimal
", "strings": ["."], "partialCredit": 0, "showStrings": false}, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "valuegenerators": [], "showPreview": true, "scripts": {}, "variableReplacements": [], "customName": "", "unitTests": [], "answer": "{2*b*c}/{2*b-a}", "variableReplacementStrategy": "originalfirst"}], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "prompt": "\nAt what value of $x \\in I$ does $g$ have a global maximum in $I$?
\n \n$x=\\;\\;$ [[0]]
\n \nAt what value of $x \\in I$ does $g$ have a global minimum in $I$ ?
\n \n$x=\\;\\;$ [[1]] (Input as a fraction or an integer and not as a decimal)
\n \n ", "unitTests": [], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true}], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variable_groups": [], "statement": "\nLet $I=[\\var{l},\\var{m}]$ be an interval and let $g: I \\rightarrow \\mathbb{R}$ be the function given by:
\\[g(x)=\\simplify{x^2/(x-{c})^({a}/{b})}\\]
Answer the following questions. There are seven parts and you may need to scroll down to complete all parts.
\n\n ", "metadata": {"description": "
$I$ compact interval. $\\displaystyle g: I\\rightarrow I, g(x)=\\frac{x^2}{(x-c)^{a/b}}$. Are there stationary points and local maxima, minima? Has $g$ a global max, global min?
", "licence": "Creative Commons Attribution 4.0 International"}, "preamble": {"css": "", "js": ""}, "functions": {}, "tags": [], "ungrouped_variables": ["a", "c", "b", "valmin", "m", "l", "xma", "lmi", "s", "u", "valend", "tol", "l1", "valbegin", "lma", "statpoint", "xmi", "valsd"], "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/"}]}