3.4.E: Second-Order Approximations (Exercises)

Exercise \(\PageIndex{1}\)

Let \(f(x, y)=x^{3} y^{2}-4 x^{2} e^{-3 y}\). Find the following.

(a) \(\frac{\partial^{2}}{\partial x \partial y} f(x, y)\)

(b) \(\frac{\partial^{2}}{\partial y \partial x} f(x, y)\)

(c) \(\frac{\partial^{2}}{\partial x^{2}} f(x, y)\)

(d) \(\frac{\partial^{3}}{\partial x \partial y \partial x} f(x, y)\)

(e) \(\frac{\partial^{3}}{\partial x \partial y^{2}} f(x, y)\)

(f) \(\frac{\partial^{3}}{\partial y^{3}} f(x, y)\)

(g) \(f_{y y}(x, y)\)

(h) \(f_{y x y}(x, y)\)

Answer

(a) \(\frac{\partial^{2}}{\partial x \partial y} f(x, y)=6 x^{2}+24 e^{-3 y}\)

(c) \(\frac{\partial^{2}}{\partial x^{2}} f(x, y)=6 x y^{2}-8 e^{-3 y}\)

(e) \(\frac{\partial^{3}}{\partial x \partial y^{2}} f(x, y)=6 x^{2}-72 x e^{-3 y}\)

(g) \(f_{y y}(x, y)=x^{3}-36 x^{2} e^{-3 y}\)

Exercise \(\PageIndex{2}\)

Let \(f(x, y, z)=\frac{x y}{x^{2}+y^{2}+z^{2}}\). Find the following.

(a) \(\frac{\partial^{2}}{\partial z \partial x} f(x, y, z)\)

(b) \(\frac{\partial^{2}}{\partial y \partial z} f(x, y, z)\)

(c) \(\frac{\partial^{2}}{\partial z^{2}} f(x, y, z)\)

(d) \(\frac{\partial^{3}}{\partial x \partial y \partial z} f(x, y, z)\)

(e) \(f_{z y x}(x, y, z)\)

(f) \(f_{y y y}(x, y, z)\)

Answer

(a) \(\frac{\partial^{2}}{\partial z \partial x} f(x, y, z)=\frac{2 y z\left(3 x^{2}-y^{2}-z^{2}\right)}{\left(x^{2}+y^{2}+z^{2}\right)^{3}}\)

(c) \(\frac{\partial^{2}}{\partial z^{2}} f(x, y, z)=\frac{2 x y\left(3 z^{2}-x^{2}-y^{2}\right)}{\left(x^{2}+y^{2}+z^{2}\right)^{3}}\)

(e) \(\frac{\partial^{3}}{\partial z \partial y \partial x} f(x, y, z)=\frac{2 z\left(3 x^{4}-18 x^{2} y^{2}+3 y^{4}+2 x^{2} z^{2}+2 y^{2} z^{2}-z^{4}\right)}{\left(x^{2}+y^{2}+z^{2}\right)^{4}}\)

Exercise \(\PageIndex{3}\)

Find the Hessian of each of the following functions.

(a) \(f(x, y)=3 x^{2} y-4 x y^{3}\)

(b) \(g(x, y)=4 e^{-x} \cos (3 y)\)

(c) \(g(x, y, z)=4 x y^{2} z^{3}\)

(d) \(f(x, y, z)=-\log \left(x^{2}+y^{2}+z^{2}\right)\)

Answer

(a) \(H f(x, y)=\left[\begin{array}{cc}
6 y & 6 x-12 y^{2} \\
6 x-12 y^{2} & -24 x y
\end{array}\right]\)

(b) \(H f(x, y, z)=\left[\begin{array}{ccc}
0 & 8 y z^{3} & 12 y^{2} z^{2} \\
8 y z^{3} & 8 x z^{3} & 24 x y z^{2} \\
12 y^{2} z^{2} & 24 x y z^{2} & 24 x y^{2} z
\end{array}\right]\)

Exercise \(\PageIndex{4}\)

Find the second-order Taylor polynomial for each of the following at the point \(\mathbf{c}\).

(a) \(f(x, y)=x e^{-y}, \mathbf{c}=(0,0)\)

(b) \(g(x, y)=x \sin (x+y), \mathbf{c}=(0,0)\)

(c) \(f(x, y)=\frac{1}{x+y}, \mathbf{c}=(1,1)\)

(d) \(g(x, y, z)=e^{x-2 y+3 z}, \mathbf{c}=(0,0,0)\)

Answer

(a) \(P_{2}(x, y)=x-x y\)

(c) \( \text { (c) } P_{2}(x, y)=\frac{1}{2}-\frac{1}{4}(x-1)-\frac{1}{4}(y-1)+\frac{1}{8}(x-1)^{2}+\frac{1}{4}(x-1)(y-1)+\frac{1}{8}(y-1)^{2}\)

Exercise \(\PageIndex{5}\)

Classify each of the following symmetric \(2 \times 2\) matrices as either positive definite, negative definite, indefinite, or nondefinite.

(a) \(\left[\begin{array}{ll}
3 & 2 \\
2 & 4
\end{array}\right]\)

(b) \(\left[\begin{array}{ll}
1 & 2 \\
2 & 2
\end{array}\right]\)

(c) \(\left[\begin{array}{rr}
-2 & 3 \\
3 & -5
\end{array}\right]\)

(d) \(\left[\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right]\)

(e) \(\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]\)

(f) \(\left[\begin{array}{ll}
8 & 4 \\
4 & 2
\end{array}\right]\)

Answer

(a) Positive definite

(c) Negative definite

(e) Positive definite

Exercise \(\PageIndex{6}\)

Let \(M\) be an \(n \times n\) symmetric nondefinite matrix and define \(q: \mathbb{R}^{n} \rightarrow \mathbb{R}\) by

\[ q(\mathbf{x})=\mathbf{x}^{T} M \mathbf{x} . \nonumber \]

Explain why (1) there exists a vector \(\mathbf{a} \neq \mathbf{0}\) such that \(q(\mathbf{a})=0\) and (2) either \(q(\mathbf{x}) \geq 0\) for all \(\mathbf{x}\) in \(\mathbb{R}^n\) or \(q(\mathbf{x}) \leq 0\) for all \(\mathbf{x}\) in \(\mathbb{R}^n\).

Exercise \(\PageIndex{7}\)

Suppose \(f: \mathbb{R}^{n} \rightarrow \mathbb{R}\) is \(C^2\) on an open ball \(B^{n}(\mathbf{c}, r), \nabla f(\mathbf{c})=\mathbf{0},\) and \(H f(\mathbf{x})\) is positive definite for all \(\mathbf{x}\) in \(B^{n}(\mathbf{c}, r)\). Show that \(f(\mathbf{c})<f(\mathbf{x})\) for all \(\mathbf{x}\) in \(B^{n}(\mathbf{c}, r)\). What would happen if \(H f(\mathbf{x})\) were negative definite for all \(\mathbf{x}\) in \(B^{n}(\mathbf{c}, r)\)? What does this say in the case \(n=1\)?

Exercise \(\PageIndex{8}\)

Let

\[ f(x, y)= \begin{cases}\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}, & \text { if }(x, y) \neq(0,0), \\ 0, & \text { if }(x, y)=(0,0). \end{cases} \nonumber \]

(a) Show that \(f_{x}(0, y)=-y\) for all \(y\).

(b) Show that \(f_{y}(x, 0)=x\) for all \(x\).

(c) Show that \(f_{y x}(0,0) \neq f_{x y}(0,0) .\)

(d) Is \(f\) \(C^2\)?