// Numbas version: exam_results_page_options {"name": "Robert's copy of Functions of two variables: Stationary points 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["rawstatval", "statval", "sol", "lmin", "q1", "dvalue", "s3", "s2", "s1", "s5", "s4", "neither", "b", "c1", "a", "c", "p1", "d", "f", "lmax", "y", "x"], "name": "Robert's copy of Functions of two variables: Stationary points 1", "tags": ["Calculus", "Differentiation", "calculus", "derivative", "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"], "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 $y$:

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\$\\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:

\n

\$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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Input both cooordinates as fractions or integers and not decimals.

\n

$x$–coordinate, $a=$ [[0]].

\n

$y$–coordinate, $b=$ [[1]].

\n

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

\n

$f(a,b)=\\;\\;$[[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.

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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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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": {}}], "type": "gapfill"}], "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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10/07/2012:

\n \t\t

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