In the following question find the $(x,y)$ coordinates of the single stationary point $(a,b)$ of the function
\n\\[f(x,y)=\\simplify[std]{{a}*x^2+{b}*x*y+{c1}*y^2+{d}*x+{f}*y}\\]
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", "partialCredit": 0}}, {"checkVariableNames": false, "answer": "{2*a*f-b*d}/{b^2-4*a*c1}", "showPreview": true, "showCorrectAnswer": true, "vsetRange": [0, 1], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "unitTests": [], "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "vsetRangePoints": 5, "scripts": {}, "checkingType": "absdiff", "marks": 2, "customMarkingAlgorithm": "", "expectedVariableNames": [], "type": "jme", "variableReplacements": [], "answerSimplification": "std", "failureRate": 1, "notallowed": {"showStrings": false, "strings": ["."], "message": "Input answer as a fraction or an integer, not a decimal
", "partialCredit": 0}}, {"mustBeReducedPC": 0, "showCorrectAnswer": true, "allowFractions": false, "type": "numberentry", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "unitTests": [], "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "scripts": {}, "correctAnswerFraction": false, "marks": 1, "customMarkingAlgorithm": "", "maxValue": "statval+0.01", "minValue": "statval-0.01", "variableReplacements": [], "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"]}], "variableReplacementStrategy": "originalfirst", "steps": [{"marks": 0, "variableReplacementStrategy": "originalfirst", "prompt": "\n \n \nThe $(x,y)$ coordinates of the stationary point of a function of 2 variables $f(x,y)$ are given by solving
the following 2 equations for $x$ and $y$
\\[\\begin{eqnarray*}\n \n \\partial f \\over \\partial x &=&0\\\\\n \n \\\\\n \n \\partial f \\over \\partial y &=&0\n \n \\end{eqnarray*}\n \n \\]
\n \n \n \nIn this case you get two linear equations to solve for $x$ and $y$
\n \n \n ", "type": "information", "variableReplacements": [], "showFeedbackIcon": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "unitTests": [], "scripts": {}}], "prompt": "Input both cooordinates as fractions or integers and not decimals.
\n$x$–coordinate, $a=$ [[0]].
\n$y$–coordinate, $b=$ [[1]].
\nInput value of $f(x,y)$ at $(a,b)$:
\n$f(a,b)=\\;\\;$[[2]] (Input to 2 decimal places).
\nIf you want some help, click on Show steps. You will not lose any marks if you do so.
", "marks": 0, "customMarkingAlgorithm": "", "scripts": {}, "type": "gapfill", "variableReplacements": [], "sortAnswers": false}], "name": "Simon's copy of Functions of two variables: Stationary points 1", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "functions": {}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "extensions": [], "variables": {"s3": {"group": "Ungrouped variables", "templateType": "anything", "name": "s3", "definition": "random(1,-1)", "description": ""}, "neither": {"group": "Ungrouped variables", "templateType": "anything", "name": "neither", "definition": "'Saddle point'", "description": ""}, "lmin": {"group": "Ungrouped variables", "templateType": "anything", "name": "lmin", "definition": "'Local minimum'", "description": ""}, "p1": {"group": "Ungrouped variables", "templateType": "anything", "name": "p1", "definition": "random(-3..3)", "description": ""}, "rawstatval": {"group": "Ungrouped variables", "templateType": "anything", "name": "rawstatval", "definition": "a*x^2+b*x*y+c1*y^2+d*x+f*y", "description": ""}, "statval": {"group": "Ungrouped variables", "templateType": "anything", "name": "statval", "definition": "precround(rawstatval,2)", "description": ""}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "name": "s2", "definition": "random(1,-1)", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "name": "b", "definition": "s2*random(1..5)", "description": ""}, "q1": {"group": "Ungrouped variables", "templateType": "anything", "name": "q1", "definition": "random(-3..3)", "description": ""}, "sol": {"group": "Ungrouped variables", "templateType": "anything", "name": "sol", "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": ""}, "d": {"group": "Ungrouped variables", "templateType": "anything", "name": "d", "definition": "s4*random(1..5)", "description": ""}, "dvalue": {"group": "Ungrouped variables", "templateType": "anything", "name": "dvalue", "definition": "4*a*c1-b^2", "description": ""}, "y": {"group": "Ungrouped variables", "templateType": "anything", "name": "y", "definition": "(2*a*f-b*d)/(b^2-4*a*c1)", "description": ""}, "s4": {"group": "Ungrouped variables", "templateType": "anything", "name": "s4", "definition": "random(1,-1)", "description": ""}, "f": {"group": "Ungrouped variables", "templateType": "anything", "name": "f", "definition": "s5*random(1..5)", "description": ""}, "x": {"group": "Ungrouped variables", "templateType": "anything", "name": "x", "definition": "(2*c1*d-b*f)/(b^2-4*a*c1)", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "name": "a", "definition": "s1*random(1..5)", "description": ""}, "s1": {"group": "Ungrouped variables", "templateType": "anything", "name": "s1", "definition": "random(1,-1)", "description": ""}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "name": "c1", "definition": "if(b^2=4*a*c,c+1,c)", "description": ""}, "s5": {"group": "Ungrouped variables", "templateType": "anything", "name": "s5", "definition": "random(1,-1)", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "name": "c", "definition": "s3*random(1..5)", "description": ""}, "lmax": {"group": "Ungrouped variables", "templateType": "anything", "name": "lmax", "definition": "'Local maximum'", "description": ""}}, "ungrouped_variables": ["rawstatval", "statval", "sol", "lmin", "q1", "dvalue", "s3", "s2", "s1", "s5", "s4", "neither", "b", "c1", "a", "c", "p1", "d", "f", "lmax", "y", "x"], "tags": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Find the stationary point $(p,q)$ of the function: $f(x,y)=ax^2+bxy+cy^2+dx+gy$. Calculate $f(p,q)$.
"}, "advice": "The $(x,y)$ coordinates of the stationary point of a function of 2 variables $f(x,y)$ are given by solving
the following 2 equations for $x$ and $y$:
\\[\\begin{eqnarray*} \\partial f \\over \\partial x &=&0\\\\ \\\\ \\partial f \\over \\partial y &=&0 \\end{eqnarray*} \\]
\nRemember that:
\n- to find $\\partial f \\over \\partial x$ we differentiate with respect to $x$, treating $y$ as a constant.
\n- to find $\\partial f \\over \\partial y$ we differentiate with respect to $y$, treating $x$ as a constant.
\n\nHence we obtain:
\n$\\partial f \\over \\partial x$ $=\\simplify[std]{{2*a}x+{b}y+{d}}$
\n$\\partial f \\over \\partial y$ $= \\simplify[std]{{b}x+{2*c1}y+{f}}$
\n\nTo find stationary points, we set each of these equations to be equal to 0, then solve simultaneously for $x$ and $y$:
\n\\[\\begin{eqnarray*} \\simplify[std]{{2*a}x+{b}y+{d}}&=&0\\\\ \\\\ \\simplify[std]{{b}x+{2*c1}y+{f}}&=&0 \\end{eqnarray*} \\]
\n\nOn 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}}\\]
On substituting these values into $f(x,y)$ we get:
\\[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{dpformat(statval,2)}\\]
to 2 decimal places.