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"vsetrange": [0, 1]}], "marks": 0, "scripts": {}, "prompt": "<p>$f(x)=\\simplify[all,!collectNumbers,!noleadingminus]{{a}x^3+{b}x^2+{c}x+{d}}$</p>\n<p>$f'(x)=$&nbsp;[[2]]</p>\n<p>$f''(x)=$&nbsp;[[3]]</p>\n<p></p>\n<p>Find when $f'(x)=0$, hence find:</p>\n<p>$x$-coordinate of the stationary point giving a minimum $=$&nbsp;[[0]]</p>\n<p>$x$-coordinate of the stationary point giving a maximum $=$&nbsp;[[1]]</p>"}], "question_groups": [{"pickQuestions": 0, "name": "", "pickingStrategy": "all-ordered", "questions": []}], "rulesets": {"std": ["all", "fractionNumbers", "!noLeadingMinus", "!collectNumbers"]}, "variable_groups": [], "functions": {}, "variables": {"lg2": {"definition": "if(mx=0,'$\\\\gt$','$\\\\lt$')", "name": "lg2", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "mx": {"definition": "if(3a*r2+b<0,0,1)", "name": "mx", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "type1": {"definition": "if(mx=0, 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{"definition": "round(-(3*a*r2+r1)/2)", "name": "b", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "r2": {"definition": "random(-6..6 except 0)", "name": "r2", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "lg1": {"definition": "if(mx=0,'$\\\\lt$','$\\\\gt$')", "name": "lg1", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "type2": {"definition": "if(mx=1, 'maximum','minimum')", "name": "type2", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "d": {"definition": "random(-10..10)", "name": "d", "group": "Ungrouped variables", "description": "", "templateType": "anything"}}, "ungrouped_variables": ["a", "c", "b", "r1", "r2", "mn", "d", "lg2", "lg1", "type1", "mx", "type2", "f1", "f2"], "name": "Stationary points of cubics", "statement": "<p><strong>Find the coordinates of the stationary points of the function.</strong></p>", "tags": ["stationary points"], "showQuestionGroupNames": false, "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Finding the stationary points of a cubic with two turning points</p>"}, "advice": "<p>On differentiating we get $\\displaystyle \\frac{df}{dx}=\\simplify[std]{{3*a}x^2+{2*b}x+{c}}$.</p>\n<p>To find the stationary points we have to solve $\\displaystyle \\frac{df}{dx}=0$&nbsp;for $x$.</p>\n<p>So we have to solve $\\simplify[std]{{3*a}x^2+{2*b}x+{c}=0}$.</p>\n<p>Note that the quadratic factorises and the equation becomes $\\simplify[std]{({3a}x-{r1})(x-{r2})=0}$.</p>\n<p>Hence we have two stationary points: $x=\\simplify[std]{{r1}/{3a}}$ and $x=\\var{r2}$.</p>\n<p>To 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 &nbsp;the stationary points.</p>\n<p>If &nbsp;$\\displaystyle \\frac{d^2f}{dx^2} \\lt 0 $ at a stationary point then it is a MAXIMUM.</p>\n<p>If &nbsp;$\\displaystyle \\frac{d^2f}{dx^2} \\gt 0 $ at a stationary point then it is a MINIMUM.</p>\n<p>If &nbsp;$\\displaystyle \\frac{d^2f}{dx^2} = 0 $ at a stationary point then we have to do more work!</p>\n<p>At $x=\\var{r2}$ we have $\\displaystyle \\frac{d^2f}{dx^2} = \\simplify{{6*a*r2+2*b}}${lg1}$0$ hence is a {type1}.</p>\n<p>At $\\displaystyle x=\\simplify[std]{{r1}/{3a}}$ we have $\\displaystyle \\frac{d^2f}{dx^2} = \\simplify{{2*r1+2*b}}${lg2}$0$ hence is a {type2}.</p>\n<p>&nbsp;</p>", "preamble": {"js": "", "css": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "contributors": [{"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}, {"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}, {"name": "Ian Davies", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3119/"}]}]}], "contributors": [{"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}, {"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}, {"name": "Ian Davies", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3119/"}]}