// Numbas version: exam_results_page_options {"name": "Find partial derivatives of $f(x,y)$ and identify its stationary points", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "ch<>0\nand\ns5=-1", "maxRuns": 100}, "variables": {"ch": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(a1*d1=b1*c1,0,1)", "name": "ch", "description": ""}, "s5": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(d*(-b*d1+4*c*c1)<=0,-1,1)", "name": "s5", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "b1*c1+a1*d1", "name": "b", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "b1*d1", "name": "c", "description": ""}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2,4,6)", "name": "b1", "description": ""}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..5)", "name": "a1", "description": ""}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2,4,6)", "name": "d1", "description": ""}, "c2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..4)", "name": "c2", "description": ""}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(b1*c2=3*a1*d1,c2+1,c2)", "name": "c1", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a1*c1", "name": "a", "description": ""}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "-(b1*c1-3*a1*d1)/2*m^2", "name": "d", "description": ""}, "m": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..4)", "name": "m", "description": ""}}, "ungrouped_variables": ["a", "c", "ch", "d", "m", "s5", "a1", "b", "b1", "c2", "c1", "d1"], "name": "Find partial derivatives of $f(x,y)$ and identify its stationary points", "functions": {}, "variable_groups": [], "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

Find the partial derivatives of $f$ with respect to $x$ and $y$.

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Note that if you want to enter a product of two unknowns, such as $xy$, then you input the expression in the form x*y.

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$\\displaystyle { \\partial f \\over \\partial x} = $ [[0]]

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$\\displaystyle {\\partial f \\over \\partial y} = $ [[1]]

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Finding Stationary Points.

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Tick the two choices which give stationary points for $f(x,y)$.

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Note that the easiest way to do this question is to substitute the values for $x$ and for $y$ into the expressions for $\\displaystyle {\\partial f \\over \\partial x}$ and $\\displaystyle{\\partial f \\over \\partial y}$ and see if you get $0$ for both.

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$x=\\var{m},\\;\\;y=\\simplify[std]{-{a1*m}/{b1}}$

", "

$x=\\var{-m},\\;\\;y=\\simplify[std]{{a1*m}/{b1}}$

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$x=\\var{m+1},\\;\\;y=\\simplify[std]{-{c1*(m+1)}/{d1}}$

", "

$x=\\var{-m-1},\\;\\;y=\\simplify[std]{{c1*(m+1)}/{d1}}$

", "

$x=\\var{m-1},\\;\\;y=\\simplify[std]{-{a1+2*b1}/{b1}}$

", "

$x=\\var{-m+1},\\;\\;y=\\simplify[std]{{a1+2*b1}/{b1}}$

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Answer the following questions about the function:

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\\[f(x,y)=\\simplify[std]{ ({a} / 3) * x ^ 3 + ({b} / 2) * x ^ 2 * y + {c} * y ^ 2 * x + {d} * y}\\]

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Find the stationary points of the function: $f(x,y)=a x ^ 3 + b x ^ 2 y + c y ^ 2 x + dy$ by choosing from a list of points.

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Inputting the values given into the partial derivatives to see if 0 is obtained is tedious! Could ask for the factorisation of equation 1 as the solution uses this. However there is a problem in asking for the input of the stationary points - order of input and also giving that there is two stationary points.

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a)

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\\begin{align}
\\partial f \\over \\partial x &= \\simplify[std]{(({a} * (x ^ 2)) + ({b} * x * y) + ({c} * (y ^ 2)))} \\\\[1em]
\\partial f \\over \\partial y &= \\simplify[std]{((({b} / 2) * (x ^ 2)) + ({(2 * c)} * x * y) + {d})}
\\end{align}

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b)

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$(a,b)$ is a stationary point for the function $f(x,y)$ if $f_x=0$ and $f_y=0$, where the partial derivatives are evaluated at $x=a$, $y=b$.

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So you have to make sure that both of these partial derivatives are $0$ at the stationary point.

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For this example we have from the above equations that:

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\\begin{align}
\\simplify[std]{(({a} * (x ^ 2)) + ({b} * x * y) + ({c} * (y ^ 2)))} &= 0, & \\mathbf{(1)}\\\\
\\simplify[std]{((({b} / 2) * (x ^ 2)) + ({(2 * c)} * x * y) + {d})} &= 0, & \\mathbf{(2)}
\\end{align}

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The left hand side of equation (1) can be factorised as:

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\\[ \\simplify[std]{({a1}x+{b1}y)*({c1}x+{d1}y)=0} \\]

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and so we have:

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\\[ y=\\simplify[std]{{-a1}/{b1}*x}, \\text{ or } y= \\simplify[std]{{-c1}/{d1}*x} \\]

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First case: $y= \\simplify[std]{{-a1}/{b1}*x}$

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Substituting this into equation (2) gives:

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\\[\\simplify[std]{{b}/2*x^2-{2c*a1}/{b1}*x^2+{d}}=0 \\implies \\simplify[std]{{-b*b1+4*c*a1}/{2*b1}*x^2={d}}\\]

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Hence $x=\\var{m}$ or $x = \\var{-m}$. The corresponding stationary points are:

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\\[ \\left(\\var{m},\\simplify[std]{-{a1*m}/{b1}}\\right) \\text{ and } \\left(\\var{-m},\\simplify[std]{{a1*m}/{b1}}\\right) \\]

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Second case: $y= \\simplify[std]{{-c1}/{d1}*x}$

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Substituting this into equation (2) gives:

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\\[\\simplify[std]{{b}/2*x^2-{2c*c1}/{d1}*x^2+{d}}=0 \\Rightarrow \\simplify[std]{{-b*d1+4*c*c1}/{2*d1}*x^2={d}}\\]

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There can be no more stationary points as this equation has no real solution.

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