// Numbas version: finer_feedback_settings
{"name": "Shaheen's copy of Functions of two variables: Locate stationary point", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Shaheen's copy of Functions of two variables: Locate stationary point", "tags": ["Calculus", "calculus", "derivative", "differentiation", "Differentiation", "function of 2 variables", "functions of 2 variables", "functions of two variables", "partial derivatives", "partial differentiation", "stationary points", "stationary points of functions of two variables"], "metadata": {"description": "<p>Find the stationary point $(p,q)$ of the function:&nbsp;$f(x,y)=ax^2+bxy+cy^2+dx+gy$. Calculate $f(p,q)$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>In the following question find the $(x,y)$ coordinates of the single stationary point $(a,b)$ of the function</p>\n<p>\\[f(x,y)=\\simplify[std]{{a}*x^2+{b}*x*y+{c1}*y^2+{d}*x+{f}*y}\\]</p>", "advice": "<p>The $(x,y)$ coordinates of the stationary point of a function of 2 variables $f(x,y)$ are given by solving<br/>the following 2 equations for $x$ and $y$:</p>\n<p>\\[\\begin{eqnarray*} \\partial f \\over \\partial x &amp;=&amp;0\\\\ \\\\ \\partial f \\over \\partial y &amp;=&amp;0 \\end{eqnarray*} \\]</p>\n<p>In this case you get two linear equations to solve for $x$ and $y$:</p>\n<p>\\[\\begin{eqnarray*} \\simplify[std]{{2*a}x+{b}y+{d}}&amp;=&amp;0\\\\ \\\\ \\simplify[std]{{b}x+{2*c1}y+{f}}&amp;=&amp;0 \\end{eqnarray*} \\]<br/>On solving these we get \\[ x = \\simplify[std]{{2*c1*d-b*f}/{b^2-4*a*c1}},\\;\\;\\;y=\\simplify[std]{{2*a*f-b*d}/{b^2-4*a*c1}}\\]<br/>On substituting these values into $f(x,y)$ we get:</p>\n<p>\\[f\\left(\\simplify[std]{{2*c1*d-b*f}/{b^2-4*a*c1}},\\simplify[std]{{2*a*f-b*d}/{b^2-4*a*c1}}\\right) = \\var{rawstatval}\\approx\\var{statval}\\]<br/>to 2 decimal places.</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"rawstatval": {"name": "rawstatval", "group": "Ungrouped variables", "definition": "a*x^2+b*x*y+c1*y^2+d*x+f*y", "description": "", "templateType": "anything", "can_override": false}, "statval": {"name": "statval", "group": "Ungrouped variables", "definition": "precround(rawstatval,2)", "description": "", "templateType": "anything", "can_override": false}, "sol": {"name": "sol", "group": "Ungrouped variables", "definition": "switch(b=1,'by multiplying the second equation by '+2*a,b=-1,'by multiplying the second equation by '+2*a,'by multiplying the second equation by '+2*a+' and multiplying the first equation by '+b)", "description": "", "templateType": "anything", "can_override": false}, "lmin": {"name": "lmin", "group": "Ungrouped variables", "definition": "'Local minimum'", "description": "", "templateType": "anything", "can_override": false}, "q1": {"name": "q1", "group": "Ungrouped variables", "definition": "random(-3..3)", "description": "", "templateType": "anything", "can_override": false}, "dvalue": {"name": "dvalue", "group": "Ungrouped variables", "definition": "4*a*c1-b^2", "description": "", "templateType": "anything", "can_override": false}, "s3": {"name": "s3", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "s2": {"name": "s2", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "s5": {"name": "s5", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "s4": {"name": "s4", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "neither": {"name": "neither", "group": "Ungrouped variables", "definition": "'Saddle point'", "description": "", "templateType": "anything", "can_override": false}, "p1": {"name": "p1", "group": "Ungrouped variables", "definition": "random(-3..3)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "if(b^2=4*a*c,c+1,c)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "s1*random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "s3*random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "s2*random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "s4*random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "s5*random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "lmax": {"name": "lmax", "group": "Ungrouped variables", "definition": "'Local maximum'", "description": "", "templateType": "anything", "can_override": false}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "(2*a*f-b*d)/(b^2-4*a*c1)", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "(2*c1*d-b*f)/(b^2-4*a*c1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["rawstatval", "statval", "sol", "lmin", "q1", "dvalue", "s3", "s2", "s1", "s5", "s4", "neither", "b", "c1", "a", "c", "p1", "d", "f", "lmax", "y", "x"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Input both cooordinates as fractions or integers and not decimals.</p>\n<p>$x$&ndash;coordinate, $a=$&nbsp;[[0]].</p>\n<p>$y$&ndash;coordinate, 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"penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}], "resources": []}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}]}