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Consider the function $f: \\mathbb{R}^3 \\longrightarrow \\mathbb{R}$ defined by $f(x,y,z) = \\simplify{{a}*x^2 + {b}*y^2 + {c}*z^2 + {d}*x*y*z}$ for all $(x,y,z) \\in \\mathbb{R}^3$.

What are the partial derivatives of $f$? (If you wish to write something like $2x$ or $2xy$, then please write 2*x (not 2x) or 2*x*y (not 2xy), respectively, so that the system correctly interprets what you wish to write).

$f_x (x,y,z) =$ [[0]] ;
$f_y (x,y,z) =$ [[1]] ;
$f_z (x,y,z) =$ [[2]] .

In this question, when we are working with $\\mathbb{R}^4$ we will use the letters $x,y,z,u$ for the four axes. For example, the horizontal plane in $\\mathbb{R}^4$ is defined by $u = 0$; and the graph of $f$ in $\\mathbb{R}^4$ is defined by $u = f(x,y,z)$, i.e. $ u = \\simplify{{a}*x^2 + {b}*y^2 + {c}*z^2 + {d}*x*y*z}$.

In this part of the question, we will find all points $(x,y,z)$ for which the tangent hyperplane to graph of $f$ at $\\big( x,y,z,f(x,y,z) \\big)$ is horizontal (i.e. parallel to the hyperplane defined by $u=0$).

This occurs when all the partial derivatives are equal to zero; i.e. $f_x (x,y,z) =0$ , $f_y (x,y,z) =0$ , and $f_z (x,y,z) =0$. Note we already calculated all the partial derivatives in part a. Rearrange $f_x (x,y,z) =0$ to express $x$ in terms of $y$ and $z$. (For the next three gaps, if you wish to write something like $\\frac{2}{3} x$, $\\frac{2}{3} xy$ or $\\frac{2}{3} x^2 y$ then please write (2/3)*x, (2/3)*x*y or (2/3)*(x^2)*y, respectively, so that the system correctly interprets what you wish to write).
$x =$ [[0]]

Now we substitute this into $f_x (x,y,z) =0$ , and$f_x (x,y,z) =0$ to get
[[1]] = 0 (euqation 1);
[[2]] = 0 (equation 2).

By factorising equation 1, we see that $y =$ [[3]] , $z =$ [[4]] , or $z =$ [[5]] . (Please enter the smallest $z$ value first).

By factorising equation 2, we see that $y =$ [[6]] , $y =$ [[7]] , or $z =$ [[8]] . (Please enter the smallest $y$ value first).

Hence we can see that the only points where all partial derivatives are zero (i.e. the only points where the tangent hyperplane is horizontal) are
$\\Big( x_1$ , [[3]] , [[8]] , $\\Big)$ ;
$\\Big( x_2$ , [[6]] , [[4]] , $\\Big)$ ;
$\\Big( x_3$ , [[6]] , [[5]] , $\\Big)$ ;
$\\Big( x_4$ , [[7]] , [[4]] , $\\Big)$ ;
$\\Big( x_5$ , [[7]] , [[5]] , $\\Big)$ .

where $x_1 =$ [[9]] , $x_2 =$ [[10]] , $x_3 =$ [[11]] , $x_4 =$ [[12]] , $x_5 =$ [[13]] .

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Now we wish to find the directional derivative of $f$ at the point $\\boldsymbol{a} = \\var{{vectorA}}$ in the direction $\\boldsymbol{u} = \\var{{vectorU}}$. That is, we wish to calculate $f_{\\boldsymbol{u}} (\\boldsymbol{a}) = \\Big\\langle \\nabla f (\\boldsymbol{a}) , \\frac{\\boldsymbol{u}}{\\| \\boldsymbol{u} \\|} \\Big\\rangle$.

First we calculate $f_{\\boldsymbol{u}} (\\boldsymbol{a}) = \\begin{pmatrix} f_x (\\boldsymbol{a}) \\\\ f_y (\\boldsymbol{a}) \\\\ f_z (\\boldsymbol{a}) \\end{pmatrix} =$ [[0]] .

Now we calculate $\\| \\boldsymbol{u} \\| =$ [[1]] .

Hence, $f_{\\boldsymbol{u}} (\\boldsymbol{a}) = \\Big\\langle \\nabla f (\\boldsymbol{a}) , \\frac{\\boldsymbol{u}}{\\| \\boldsymbol{u} \\|} \\Big\\rangle =$ [[2]] .

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For the remainder of the question we will look at classifying stationary points using principle minors. This is covered in lecture 32.

i) Consider the function $h: \\mathbb{R}^3 \\longrightarrow \\mathbb{R}$ defined by $h(x,y,z) = \\simplify{{u}*x - y*x + (1/2)*y^2 - {t}*y + {q1}*z^2 - 2*{m1}*{q1}*z}$ for all $(x,y,z) \\in \\mathbb{R}^3$.

Calculate the stationary point, $\\boldsymbol{x_1}$, of this function. This is similar, but easier, than what we did in part b.
$\\boldsymbol{x_1} =$ [[0]] .

Now determine the second derivate of $h$ at $\\boldsymbol{x_1}$.
$h'' (\\boldsymbol{x_1}) =$ [[1]] .

We can see that the determinant of $h'' (\\boldsymbol{x_1})$ (which is also the third principle minor) is [[2]] . This is non-zero and so the principle minors test will certainly work. The first principle minor is [[3]] and the second principle minor is [[4]] .

Hence, $h'' (\\boldsymbol{x_1})$ is [[5]] and so our stationary point is a [[6]] .

ii) Consider the function $h: \\mathbb{R}^3 \\longrightarrow \\mathbb{R}$ defined by $h(x,y,z) = \\simplify{{q2}*x^2 + {r2}*y^2 + {s2}*z^2 - 2*{m2}*{q2}*x - 2*{n2}*{r2}*y - 2*{p2}*{s2}*z}$ for all $(x,y,z) \\in \\mathbb{R}^3$.

Calculate the stationary point, $\\boldsymbol{x_2}$, of this function. This is similar, but easier, than what we did in part b.
$\\boldsymbol{x_2} =$ [[7]] .

Now determine the second derivate of $h$ at $\\boldsymbol{x_2}$.
$h'' (\\boldsymbol{x_2}) =$ [[8]] .

We can see that the determinant of $h'' (\\boldsymbol{x_2})$ (which is also the third principle minor) is [[9]] . This is non-zero and so the principle minors test will certainly work. The first principle minor is [[10]] and the second principle minor is [[11]] .

Hence, $h'' (\\boldsymbol{x_2})$ is [[12]] and so our stationary point is a [[13]] .

iii) Consider the function $h: \\mathbb{R}^3 \\longrightarrow \\mathbb{R}$ defined by $h(x,y,z) = \\simplify{{q3}*x^2 + {r3}*y^2 + {s3}*z^2 - 2*{m3}*{q3}*x - 2*{n3}*{r3}*y - 2*{p3}*{s3}*z}$ for all $(x,y,z) \\in \\mathbb{R}^3$.

Calculate the stationary point, $\\boldsymbol{x_3}$, of this function. This is similar, but easier, than what we did in part b.
$\\boldsymbol{x_3} =$ [[14]] .

Now determine the second derivate of $h$ at $\\boldsymbol{x_3}$.
$h'' (\\boldsymbol{x_3}) =$ [[15]] .

We can see that the determinant of $h'' (\\boldsymbol{x_3})$ (which is also the third principle minor) is [[16]] . This is non-zero and so the principle minors test will certainly work. The first principle minor is [[17]] and the second principle minor is [[18]] .

Hence, $h'' (\\boldsymbol{x_3})$ is [[19]] and so our stationary point is a [[20]] .

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negative definite

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saddle point

", "

maximum point

", "

minimum point

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This helps form part of our function in part d)i), namely the coefficients.

", "definition": "random(1..5)"}, "q3": {"name": "q3", "templateType": "anything", "group": "Variables for part d subpart iii", "description": "

This helps form part of our function in part d)iii), namely the coefficients.

", "definition": "random(-5..-1)"}, "SP1": {"name": "SP1", "templateType": "anything", "group": "Variables for part d subpart i", "description": "

This is the stationary point in part d)i).

", "definition": "vector(u - t , u , m1)"}, "Det1": {"name": "Det1", "templateType": "anything", "group": "Variables for part d subpart i", "description": "

This is the second derivative in part d)i).

", "definition": "matrix([0 , -1 , 0] , [-1 , 1 , 0] , [0 , 0 , 2*q1])"}, "SP2": {"name": "SP2", "templateType": "anything", "group": "Variables for part d subpart ii", "description": "

This is the stationary point in part d)ii).

", "definition": "vector(m2 , n2 , p2)"}, "r3": {"name": "r3", "templateType": "anything", "group": "Variables for part d subpart iii", "description": "

This helps form part of our function in part d)iii), namely the coefficients.

", "definition": "random(-5..-1)"}, "t": {"name": "t", "templateType": "anything", "group": "Variables for part d subpart i", "description": "

This helps form part of our function in part d)i), namely the coefficients.

", "definition": "random(-5..5 except(0) except(u))"}, "p2": {"name": "p2", "templateType": "anything", "group": "Variables for part d subpart ii", "description": "

This helps form part of our function in part d)ii), namely the coefficients.

", "definition": "random(-5..5 except(0))"}, "n2": {"name": "n2", "templateType": "anything", "group": "Variables for part d subpart ii", "description": "

This helps form part of our function in part d)ii), namely the coefficients.

", "definition": "random(-5..5 except(0))"}, "b": {"name": "b", "templateType": "anything", "group": "Variables for parts a to c", "description": "

This is one of the coefficients of our function defined in part a.

", "definition": "(random(1..5))^2"}, "s2": {"name": "s2", "templateType": "anything", "group": "Variables for part d subpart ii", "description": "

This helps form part of our function in part d)ii), namely the coefficients.

", "definition": "random(1..5)"}, "n3": {"name": "n3", "templateType": "anything", "group": "Variables for part d subpart iii", "description": "

This helps form part of our function in part d)iii), namely the coefficients.

", "definition": "random(-5..5 except(0))"}, "q1": {"name": "q1", "templateType": "anything", "group": "Variables for part d subpart i", "description": "

This helps form part of our function in part d)i), namely the coefficients.

", "definition": "random(1..5)"}, "p3": {"name": "p3", "templateType": "anything", "group": "Variables for part d subpart iii", "description": "

This helps form part of our function in part d)iii), namely the coefficients.

", "definition": "random(-5..5 except(0))"}, "vectorU": {"name": "vectorU", "templateType": "anything", "group": "Variables for parts a to c", "description": "

This is the direction vector in part c.

", "definition": "vector(1,2,2)"}, "nabla_f_vectorA": {"name": "nabla_f_vectorA", "templateType": "anything", "group": "Variables for parts a to c", "description": "

This is the value of nabla f at the point vectorX.

", "definition": "vector(2*{a}*{f} + {d}*{g}*{h} , 2*{b}*{g} + {d}*{f}*{h} , 2*{c}*{h} + {d}*{f}*{g})"}, "a": {"name": "a", "templateType": "anything", "group": "Variables for parts a to c", "description": "

This is one of the coefficients of our function defined in part a.

", "definition": "(random(1..5))^2"}, "m3": {"name": "m3", "templateType": "anything", "group": "Variables for part d subpart iii", "description": "

This helps form part of our function in part d)iii), namely the coefficients.

", "definition": "random(-5..5 except(0))"}, "u": {"name": "u", "templateType": "anything", "group": "Variables for part d subpart i", "description": "

This helps form part of our function in part d)i), namely the coefficients.

", "definition": "random(-5..5 except(0))"}, "m2": {"name": "m2", "templateType": "anything", "group": "Variables for part d subpart ii", "description": "

This helps form part of our function in part d)ii), namely the coefficients.

", "definition": "random(-5..5 except(0))"}, "g": {"name": "g", "templateType": "anything", "group": "Variables for parts a to c", "description": "

This is the y coordinate of our point in part c.

", "definition": "random(1..5)"}, "r2": {"name": "r2", "templateType": "anything", "group": "Variables for part d subpart ii", "description": "

This helps form part of our function in part d)ii), namely the coefficients.

", "definition": "random(1..5)"}, "f_vectorU_vectorA": {"name": "f_vectorU_vectorA", "templateType": "anything", "group": "Variables for parts a to c", "description": "

This is the directional derivative of f at the point vectorX in the direction vectorU

", "definition": "dot(nabla_f_vectorA , (1/5)*vectorU)"}, "s3": {"name": "s3", "templateType": "anything", "group": "Variables for part d subpart iii", "description": "

This helps form part of our function in part d)iii), namely the coefficients.

", "definition": "random(-5..-1)"}, "q2": {"name": "q2", "templateType": "anything", "group": "Variables for part d subpart ii", "description": "

This helps form part of our function in part d)ii), namely the coefficients.

", "definition": "random(1..5)"}, "c": {"name": "c", "templateType": "anything", "group": "Variables for parts a to c", "description": "

This is one of the coefficients of our function defined in part a. It is chosen so that it is not equal to the variable b. This ensure that we have 5 points where the tangent plane is horizontal (if c-b then we would have only 3).

", "definition": "(random(1..5 except((b)^(1/2))))^2"}, "Det2": {"name": "Det2", "templateType": "anything", "group": "Variables for part d subpart ii", "description": "

This is the second derivative in part d)ii).

", "definition": "matrix([2*q2 , 0 , 0] , [0 , 2*r2 , 0] , [0 , 0 , 2*s2])"}, "f": {"name": "f", "templateType": "anything", "group": "Variables for parts a to c", "description": "

This is the x coordinate of our point in part c.

", "definition": "random(1..5)"}, "Det3": {"name": "Det3", "templateType": "anything", "group": "Variables for part d subpart iii", "description": "

This is the second derivative in part d)iii).

", "definition": "matrix([2*q3 , 0 , 0] , [0 , 2*r3 , 0] , [0 , 0 , 2*s3])"}, "h": {"name": "h", "templateType": "anything", "group": "Variables for parts a to c", "description": "

This is the z coordinate of our point in part c. It is define to not equal the variable g as this is a sufficient (although not necessary) condition to ensure that the the gradient of f at the point (f,g,h) is not the zero vector.

", "definition": "random(1..5 except(g))"}, "vectorA": {"name": "vectorA", "templateType": "anything", "group": "Variables for parts a to c", "description": "", "definition": "vector(f,g,h)"}, "d": {"name": "d", "templateType": "anything", "group": "Variables for parts a to c", "description": "

This is one of the coefficients of our function defined in part a.

", "definition": "random(1..5)"}, "SP3": {"name": "SP3", "templateType": "anything", "group": "Variables for part d subpart iii", "description": "

This is the stationary point in part d)iii).

", "definition": "vector(m3 , n3 , p3)"}}, "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "name": "MA100 LT Week 6", "statement": "

Lent Term Week 6 (lectures 31 and 32): In this question you will look at partial derivatives, tangent hyperplanes, and classifying stationary points using principle minors.

Please read the following before attempting the question:

If you have not provided an answer to every input gap of a question or part of the question, and you try to submit your answers to the question or part, then you will see the message \"Can not submit answer - check for errors\". In reality your answer has been submitted, but the system is just concerned that you have not submitted an answer to every input gap. For this reason, please ensure that you provide an answer to every input gap in the question or part before submitting. Even if you are unsure of the answer, write down what you think is most likely to be correct; you can always change your answer or retry the question.

As with all questions, there may be parts where you can choose to \"Show steps\". This may give a hint, or it may present sub-parts which will help you to solve that part of the question. Furthermore, remember to always press the \"Show feedback\" button at the end of each part. Sometimes, helpful feedback will be given here, and often it will depend on how correctly you have answered and will link to other parts of the question. Hence, always retry the parts until you obtain full marks, and then look at the feedback again.

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This is the question for Lent Term week 6 of the MA100 course at the LSE. It looks at material from chapters 31 and 32.

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