// Numbas version: exam_results_page_options {"name": "George's copy of Functions of two variables: Stationary points 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"c": {"definition": "s3*random(1..5)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "c"}, "b": {"definition": "s2*random(1..5)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "b"}, "x": {"definition": "(2*c1*d-b*f)/(b^2-4*a*c1)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "x"}, "q1": {"definition": "random(-3..3)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "q1"}, "s3": {"definition": "random(1,-1)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "s3"}, "f": {"definition": "s5*random(1..5)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "f"}, "s2": {"definition": "random(1,-1)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "s2"}, "s1": {"definition": "random(1,-1)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "s1"}, "lmin": {"definition": "'Local minimum'", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "lmin"}, "rawstatval": {"definition": "a*x^2+b*x*y+c1*y^2+d*x+f*y", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "rawstatval"}, "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": "", "templateType": "anything", "group": "Ungrouped variables", "name": "sol"}, "dvalue": {"definition": "4*a*c1-b^2", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "dvalue"}, "s4": {"definition": "random(1,-1)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "s4"}, "a": {"definition": "s1*random(1..5)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "a"}, "d": {"definition": "s4*random(1..5)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "d"}, "neither": {"definition": "'Saddle point'", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "neither"}, "c1": {"definition": "if(b^2=4*a*c,c+1,c)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "c1"}, "statval": {"definition": "precround(rawstatval,2)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "statval"}, "lmax": {"definition": "'Local maximum'", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "lmax"}, "y": {"definition": "(2*a*f-b*d)/(b^2-4*a*c1)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "y"}, "s5": {"definition": "random(1,-1)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "s5"}, "p1": {"definition": "random(-3..3)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "p1"}}, "statement": "

In the following question find the $(x,y)$ coordinates of the single stationary point $(a,b)$ of the function

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\\[f(x,y)=\\simplify[std]{{a}*x^2+{b}*x*y+{c1}*y^2+{d}*x+{f}*y}\\]

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Input answer as a fraction or an integer, not a decimal

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Input answer as a fraction or an integer, not a decimal

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Input both cooordinates as fractions or integers and not decimals.

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$x$–coordinate, $a=$ [[0]].

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$y$–coordinate, $b=$ [[1]].

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Input value of $f(x,y)$ at $(a,b)$:

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$f(a,b)=\\;\\;$[[2]] (Input to 2 decimal places).

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If you want some help, click on Show steps. You will not lose any marks if you do so.

", "steps": [{"type": "information", "prompt": "\n \n \n

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 $y$

\n \n \n \n

\\[\\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 \n

In this case you get two linear equations to solve for $x$ and $y$

\n \n \n ", "scripts": {}, "marks": 0, "showCorrectAnswer": true}]}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "notes": "\n \t\t

10/07/2012:

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Added tags.

\n \t\t

Question appears to be working correctly.

\n \t\t", "description": "

Find the stationary point $(p,q)$ of the function: $f(x,y)=ax^2+bxy+cy^2+dx+gy$. Calculate $f(p,q)$.

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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 $y$:

\n

\\[\\begin{eqnarray*} \\partial f \\over \\partial x &=&0\\\\ \\\\ \\partial f \\over \\partial y &=&0 \\end{eqnarray*} \\]

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In this case you get two linear equations to solve for $x$ and $y$:

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\\[\\begin{eqnarray*} \\simplify[std]{{2*a}x+{b}y+{d}}&=&0\\\\ \\\\ \\simplify[std]{{b}x+{2*c1}y+{f}}&=&0 \\end{eqnarray*} \\]
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}}\\]
On substituting these values into $f(x,y)$ we get:

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\\[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}\\]
to 2 decimal places.

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