// Numbas version: exam_results_page_options
{"name": "Applications of differentiation 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"type": "question", "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>", "parts": [{"type": "gapfill", "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>", "showCorrectAnswer": true, "marks": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "scripts": {}, "gaps": [{"type": "jme", "vsetrangepoints": 5, "checkingtype": "absdiff", "answersimplification": "std", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showpreview": true, "answer": "{mx*r2}+{(1-mx)}*{(r1)}/{(3*a)}", "checkvariablenames": false, "showCorrectAnswer": true, "marks": 1, "checkingaccuracy": 0.001, "expectedvariablenames": [], "vsetrange": [0, 1], "scripts": {}}, {"type": "jme", "vsetrangepoints": 5, "checkingtype": "absdiff", "answersimplification": "std", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showpreview": true, "answer": "{mn*r2}+{(1-mn)}*{(r1)}/{(3*a)}", "checkvariablenames": false, "showCorrectAnswer": true, "marks": 1, "checkingaccuracy": 0.001, "expectedvariablenames": [], "vsetrange": [0, 1], "scripts": {}}, {"type": "jme", "vsetrangepoints": 5, "checkingtype": "absdiff", "answersimplification": "all", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showpreview": true, "answer": "3{a}x^2+2{b}x+{c}", "checkvariablenames": false, "showCorrectAnswer": true, "marks": "1", "checkingaccuracy": 0.001, "expectedvariablenames": [], "vsetrange": [0, 1], "scripts": {}}, {"type": "jme", "vsetrangepoints": 5, "checkingtype": "absdiff", "answersimplification": "all", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showpreview": true, "answer": "6{a}x+2{b}", "checkvariablenames": false, "showCorrectAnswer": true, "marks": "1", "checkingaccuracy": 0.001, "expectedvariablenames": [], "vsetrange": [0, 1], "scripts": {}}]}], "metadata": {"description": "<p>Finding the stationary points of a cubic with two turning points</p>", "notes": "", "licence": "Creative Commons Attribution 4.0 International"}, "preamble": {"js": "", "css": ""}, "variable_groups": [], "statement": "<p><strong>Find the coordinates of the stationary points of the function.</strong></p>", "rulesets": {"std": ["all", "fractionNumbers", "!noLeadingMinus", "!collectNumbers"]}, "showQuestionGroupNames": false, "name": "Applications of differentiation 2", "variablesTest": {"condition": "", "maxRuns": 100}, "functions": {}, "tags": ["stationary points"], "variables": {"lg2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(mx=0,'$\\\\gt$','$\\\\lt$')", "description": "", "name": "lg2"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "round(-(3*a*r2+r1)/2)", "description": "", "name": "b"}, "type2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(mx=1, 'maximum','minimum')", "description": "", "name": "type2"}, "mn": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(3a*r2+b<0,1,0)", "description": "", "name": "mn"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-10..10)", "description": "", "name": "d"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-2..2 except 0)", "description": "", "name": "a"}, "lg1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(mx=0,'$\\\\lt$','$\\\\gt$')", "description": "", "name": "lg1"}, "r2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "name": "r2"}, "mx": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(3a*r2+b<0,0,1)", "description": "", "name": "mx"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "r1*r2", "description": "", "name": "c"}, "f2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(-(2{b})+sqrt((2{b})^2-4(3{a}){c}))/(6{a})", "description": "", "name": "f2"}, "r1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-4..4#2 except 0)-3*a*r2", "description": "", "name": "r1"}, "f1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(-(2{b})-sqrt((2{b})^2-4(3{a}){c}))/(6{a})", "description": "", "name": "f1"}, "type1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(mx=0, 'maximum','minimum')", "description": "", "name": "type1"}}, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "questions": [], "pickQuestions": 0}], "ungrouped_variables": ["a", "c", "b", "r1", "r2", "mn", "d", "lg2", "lg1", "type1", "mx", "type2", "f1", "f2"], "contributors": [{"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}]}]}], "contributors": [{"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}]}