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{"name": "Simon's copy of Turning points", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "<p>The function \\(f(x)=2x^3-\\simplify{3*({a}+{b})x^2+6*{a}*{b}x}+\\var{c}\\) &nbsp;has two turning points.</p>", "variablesTest": {"maxRuns": 100, "condition": ""}, "advice": "<p>\\(f(x)=2x^3-\\simplify{3*({a}+{b})x^2+6*{a}*{b}x}+\\var{c}\\)</p>\n<p>To locate a turning point, we&nbsp;need to find points where the derivative $f'(x)=0$</p>\n<p>First we find $f'(x)$:</p>\n<p>\\(f'(x)=6x^2-\\simplify{6*({a}+{b})x+6*{a}*{b}}\\)</p>\n<p></p>\n<p>To find the turning points, we must set this to be equal to zero and then solve for $x$. i.e.&nbsp;</p>\n<p>Solve $6x^2-\\simplify{6*({a}+{b})x+6*{a}*{b}}=0$</p>\n<p>Divide across by 6 to get the quadratic equation</p>\n<p style=\"padding-left: 30px;\">\\(x^2-\\simplify{({a}+{b})x+{a}*{b}}=0\\)</p>\n<p>This has factors</p>\n<p style=\"padding-left: 30px;\">\\((x-\\var{a})(x-\\var{b})=0\\)</p>\n<p style=\"padding-left: 30px;\">\\(x-\\var{a}=0\\) &nbsp; &nbsp; or &nbsp; &nbsp; \\(x-\\var{b}=0\\)</p>\n<p style=\"padding-left: 30px;\">\\(x=\\var{a}\\) &nbsp; &nbsp; or &nbsp; &nbsp; \\(x=\\var{b}\\)</p>", "rulesets": {}, "variable_groups": [], "parts": [{"customMarkingAlgorithm": "", "marks": 0, "variableReplacements": [], "prompt": "<p>Input the smaller of the two \\(x\\)&nbsp;values.&nbsp;</p>\n<p style=\"padding-left: 30px;\">\\(x=\\)&nbsp;[[0]]</p>\n<p>Input the larger of the two \\(x\\) values.</p>\n<p style=\"padding-left: 30px;\">\\(x=\\)&nbsp;[[1]]</p>", "type": "gapfill", "gaps": [{"customMarkingAlgorithm": "", "correctAnswerStyle": "plain", "marks": 1, "mustBeReducedPC": 0, "allowFractions": false, "variableReplacements": [], "scripts": {}, "type": "numberentry", "showFeedbackIcon": true, "unitTests": [], "notationStyles": ["plain", "en", "si-en"], "minValue": "{a}", "variableReplacementStrategy": "originalfirst", "maxValue": "{a}", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "correctAnswerFraction": false, "showCorrectAnswer": true}, {"customMarkingAlgorithm": "", "correctAnswerStyle": "plain", "marks": 1, "mustBeReducedPC": 0, "allowFractions": false, "variableReplacements": [], "scripts": {}, "type": "numberentry", "showFeedbackIcon": true, "unitTests": [], "notationStyles": ["plain", "en", "si-en"], "minValue": "{b}", "variableReplacementStrategy": "originalfirst", "maxValue": "{b}", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "correctAnswerFraction": false, "showCorrectAnswer": true}], "showFeedbackIcon": true, "unitTests": [], "scripts": {}, "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": false}], "functions": {}, "tags": [], "name": "Simon's copy of Turning points", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Turning points of a cubic function</p>"}, "variables": {"b": {"name": "b", "definition": "random(6..11#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}, "c": {"name": "c", "definition": "random(1..15#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}, "a": {"name": "a", "definition": "random(1..5#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}}, "ungrouped_variables": ["a", "b", "c"], "preamble": {"js": "", "css": ""}, "extensions": [], "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}