First differentiate:
\n\n$f'(x)=$ [[0]]
\n\n$f''(x)=$ [[1]]
\n\n", "marks": 0, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "customMarkingAlgorithm": ""}, {"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "showFeedbackIcon": true, "gaps": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "jme", "scripts": {}, "vsetRange": [0, 1], "checkingAccuracy": 0.001, "showFeedbackIcon": true, "unitTests": [], "marks": "3", "variableReplacements": [], "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "answer": "{mx*r2}+{(1-mx)}*{(r1)}/{(3*a)}", "customMarkingAlgorithm": "", "showPreview": true, "checkVariableNames": false, "answerSimplification": "all", "expectedVariableNames": [], "checkingType": "absdiff"}, {"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "jme", "scripts": {}, "vsetRange": [0, 1], "checkingAccuracy": 0.001, "showFeedbackIcon": true, "unitTests": [], "marks": "3", "variableReplacements": [], "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "answer": "{mn*r2}+{(1-mn)}*{(r1)}/{(3*a)}", "customMarkingAlgorithm": "", "showPreview": true, "checkVariableNames": false, "answerSimplification": "all", "expectedVariableNames": [], "checkingType": "absdiff"}], "unitTests": [], "prompt": "Solve the equation $f'(x)=0$, and hence find:
\n\nThe $x$-coordinate of the stationary point giving a minimum $=$ [[0]]
\n\nThe $x$-coordinate of the stationary point giving a maximum $=$ [[1]]
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\nTo find the stationary points we have to solve $\\displaystyle \\frac{df}{dx}=0$ for $x$.
\nSo we have to solve $\\simplify[std]{{3*a}x^2+{2*b}x+{c}=0}$.
\nNote that the quadratic factorises and the equation becomes $\\simplify[std]{({3a}x-{r1})(x-{r2})=0}$.
\nHence we have two stationary points: $x=\\simplify[std]{{r1}/{3a}}$ and $x=\\var{r2}$.
\nTo find out the types of these stationary points we look at the sign of $\\displaystyle \\frac{d^2f}{dx^2} = \\simplify{{6a}*x+{2*b}}$ at the stationary points.
\nIf $\\displaystyle \\frac{d^2f}{dx^2} \\lt 0 $ at a stationary point then it is a MAXIMUM.
\nIf $\\displaystyle \\frac{d^2f}{dx^2} \\gt 0 $ at a stationary point then it is a MINIMUM.
\nIf $\\displaystyle \\frac{d^2f}{dx^2} = 0 $ at a stationary point then we have to do more work!
\nAt $x=\\var{r2}$ we have $\\displaystyle \\frac{d^2f}{dx^2} = \\simplify{{6*a*r2+2*b}}${lg1}$0$ hence is a {type1}.
\nAt $\\displaystyle x=\\simplify[std]{{r1}/{3a}}$ we have $\\displaystyle \\frac{d^2f}{dx^2} = \\simplify{{2*r1+2*b}}${lg2}$0$ hence is a {type2}.
\n", "name": "Differentiation: Cubic Stationary Points", "preamble": {"css": "", "js": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "extensions": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "
Finding the stationary points of a cubic with two turning points
"}, "rulesets": {"std": ["all", "fractionNumbers", "!noLeadingMinus", "!collectNumbers"]}, "statement": "Find the coordinates of the stationary points of the function:
\n\n
$f(x)=\\simplify[all,!collectNumbers,!noleadingminus]{{a}x^3+{b}x^2+{c}x+{d}}$
\n\n", "functions": {}, "type": "question", "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}, {"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}]}]}], "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}, {"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}]}