// 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": [{"preamble": {"css": "", "js": ""}, "parts": [{"type": "gapfill", "marks": 0, "prompt": "
Input the smaller of the two \\(x\\) values.
\n\\(x=\\) [[0]]
\nInput the larger of the two \\(x\\) values.
\n\\(x=\\) [[1]]
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", "ungrouped_variables": ["a", "b", "c"], "extensions": [], "name": "Turning points", "advice": "\\(f(x)=2x^3-\\simplify{3*({a}+{b})x^2+6*{a}*{b}x}+\\var{c}\\)
\nTo locate a turning point, differentite the function, set equal to zero and solve.
\n\\(f'(x)=6x^2-\\simplify{6*({a}+{b})x+6*{a}*{b}}=0\\)
\nDivide across by 6 to get the quadratic equation
\n\\(x^2-\\simplify{({a}+{b})x+{a}*{b}}=0\\)
\nThis has factors
\n\\((x-\\var{a})(x-\\var{b})=0\\)
\n\\(x-\\var{a}=0\\) or \\(x-\\var{b}=0\\)
\n\\(x=\\var{a}\\) or \\(x=\\var{b}\\)
", "metadata": {"description": "Turning points of a cubic function
", "licence": "Creative Commons Attribution 4.0 International"}, "functions": {}, "rulesets": {}, "variable_groups": [], "variables": {"a": {"definition": "random(1..5#1)", "templateType": "randrange", "description": "", "name": "a", "group": "Ungrouped variables"}, "c": {"definition": "random(1..15#1)", "templateType": "randrange", "description": "", "name": "c", "group": "Ungrouped variables"}, "b": {"definition": "random(6..11#1)", "templateType": "randrange", "description": "", "name": "b", "group": "Ungrouped variables"}}, "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}