// 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": {"showfrontpage": false, "preventleave": false, "allowregen": true}, "question_groups": [{"questions": [{"ungrouped_variables": ["a", "c", "ch", "d", "m", "s5", "a1", "b", "b1", "c2", "c1", "d1"], "advice": "

\n

\\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} \n #### b) \n (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. \n So you have to make sure that both of these partial derivatives are 0 at the stationary point. \n For this example we have from the above equations that: \n \\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} \n The left hand side of equation (1) can be factorised as: \n \\[ \\simplify[std]{({a1}x+{b1}y)*({c1}x+{d1}y)=0} \

\n

and so we have:

\n

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

\n

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

\n

Substituting this into equation (2) gives:

\n

\$\\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}}\$

\n

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

\n

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

\n

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

\n

Substituting this into equation (2) gives:

\n

\$\\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}}\$

\n

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

", "parts": [{"variableReplacementStrategy": "originalfirst", "prompt": "

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

\n

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.

\n

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

\n

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

", "marks": 0, "type": "gapfill", "unitTests": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "sortAnswers": false, "showFeedbackIcon": true, "gaps": [{"variableReplacementStrategy": "originalfirst", "vsetRange": [0, 1], "answerSimplification": "std", "checkingType": "absdiff", "marks": 2, "showPreview": true, "type": "jme", "expectedVariableNames": ["x", "y"], "checkingAccuracy": 0.001, "unitTests": [], "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "answer": "(({a} * (x ^ 2)) + ({b} * x * y) + ({c} * (y ^ 2)))", "customMarkingAlgorithm": "", "showFeedbackIcon": true, "checkVariableNames": true, "scripts": {}, "failureRate": 1, "variableReplacements": [], "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "vsetRange": [0, 1], "answerSimplification": "std", "checkingType": "absdiff", "marks": 2, "showPreview": true, "type": "jme", "expectedVariableNames": ["x", "y"], "checkingAccuracy": 0.001, "unitTests": [], "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "answer": "((({b} / 2) * (x ^ 2)) + ({(2 * c)} * x * y) + {d})", "customMarkingAlgorithm": "", "showFeedbackIcon": true, "checkVariableNames": true, "scripts": {}, "failureRate": 1, "variableReplacements": [], "showCorrectAnswer": true}], "variableReplacements": [], "customMarkingAlgorithm": ""}, {"variableReplacementStrategy": "originalfirst", "prompt": "\n

#### Finding Stationary Points.

\n

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

\n

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.

\n ", "marks": 0, "minMarks": 0, "scripts": {}, "unitTests": [], "customMarkingAlgorithm": "", "showFeedbackIcon": true, "shuffleChoices": true, "variableReplacements": [], "displayColumns": 3, "warningType": "none", "choices": ["

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

", "

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

", "

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

", "

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

", "

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

", "

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

"], "distractors": ["", "", "", "", "", ""], "extendBaseMarkingAlgorithm": true, "type": "m_n_2", "displayType": "checkbox", "matrix": [2, 2, 0, 0, 0, 0], "showCorrectAnswer": true, "maxMarks": 0, "minAnswers": 0, "maxAnswers": 0}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "tags": ["calculus", "functions", "multivariable", "partial derivatives", "stationary points"], "type": "question", "extensions": [], "functions": {}, "metadata": {"description": "

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.

\n

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.

", "licence": "Creative Commons Attribution 4.0 International"}, "name": "Find partial derivatives of $f(x,y)$ and identify its stationary points", "variables": {"c1": {"description": "", "templateType": "anything", "name": "c1", "definition": "if(b1*c2=3*a1*d1,c2+1,c2)", "group": "Ungrouped variables"}, "b1": {"description": "", "templateType": "anything", "name": "b1", "definition": "random(2,4,6)", "group": "Ungrouped variables"}, "s5": {"description": "", "templateType": "anything", "name": "s5", "definition": "if(d*(-b*d1+4*c*c1)<=0,-1,1)", "group": "Ungrouped variables"}, "a": {"description": "", "templateType": "anything", "name": "a", "definition": "a1*c1", "group": "Ungrouped variables"}, "d": {"description": "", "templateType": "anything", "name": "d", "definition": "-(b1*c1-3*a1*d1)/2*m^2", "group": "Ungrouped variables"}, "b": {"description": "", "templateType": "anything", "name": "b", "definition": "b1*c1+a1*d1", "group": "Ungrouped variables"}, "c2": {"description": "", "templateType": "anything", "name": "c2", "definition": "random(1..4)", "group": "Ungrouped variables"}, "m": {"description": "", "templateType": "anything", "name": "m", "definition": "random(1..4)", "group": "Ungrouped variables"}, "c": {"description": "", "templateType": "anything", "name": "c", "definition": "b1*d1", "group": "Ungrouped variables"}, "ch": {"description": "", "templateType": "anything", "name": "ch", "definition": "if(a1*d1=b1*c1,0,1)", "group": "Ungrouped variables"}, "a1": {"description": "", "templateType": "anything", "name": "a1", "definition": "random(1..5)", "group": "Ungrouped variables"}, "d1": {"description": "", "templateType": "anything", "name": "d1", "definition": "random(2,4,6)", "group": "Ungrouped variables"}}, "variable_groups": [], "statement": "\n

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