// Numbas version: exam_results_page_options
{"name": "Turning points", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Turning points", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Turning points of a cubic function</p>"}, "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, differentite the function, set equal to zero and solve.</p>\n<p style=\"padding-left: 30px;\">\\(f'(x)=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>", "extensions": [], "type": "question", "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>", "variable_groups": [], "functions": {}, "variables": {"b": {"templateType": "randrange", "name": "b", "description": "", "group": "Ungrouped variables", "definition": "random(6..11#1)"}, "c": {"templateType": "randrange", "name": "c", "description": "", "group": "Ungrouped variables", "definition": "random(1..15#1)"}, "a": {"templateType": "randrange", "name": "a", "description": "", "group": "Ungrouped variables", "definition": "random(1..5#1)"}}, "tags": [], "parts": [{"variableReplacements": [], "showFeedbackIcon": true, "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>", "scripts": {}, "showCorrectAnswer": false, "type": "gapfill", "gaps": [{"variableReplacements": [], "minValue": "{a}", "type": "numberentry", "maxValue": "{a}", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "allowFractions": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1}, {"variableReplacements": [], "minValue": "{b}", "type": "numberentry", "maxValue": "{b}", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "allowFractions": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1}], "variableReplacementStrategy": "originalfirst", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"css": "", "js": ""}, "rulesets": {}, "ungrouped_variables": ["a", "b", "c"], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}