Mysterious functions

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We describe mysterious functions

Work in groups of 3–4, writing your answers on a separate sheet of paper.

Consider the differentiable function $F:\R ^2\to \R$ where: $\begin{align*} F(2,3) &= 1 & F^{(1,0)}(2,3) &= 0 & F^{(2,0)}(2,3) &= -3\\ F^{(1,1)}(2,3) &=0 & F^{(0,1)}(2,3) &= 0 & F^{(0,2)}(2,3) &= -4 \end{align*}$

(a): Write down the second degree Taylor polynomial for $F$ centered at $(2,3)$ .
(b): Is the point $(2,3,F(2,3))$ a local min, a local max, a saddle, or none of these?

Consider the differentiable function $G:\R ^2\to \R$ where: $\begin{align*} G(-2,1) &= -4 & G^{(1,0)}(-2,1) &= 0 & G^{(2,0)}(-2,1) &= 2\\ G^{(1,1)}(-2,1) &=3 & G^{(0,1)}(-2,1) &= 0 & G^{(0,2)}(-2,1) &= 5 \end{align*}$

(a): Write down the second degree Taylor polynomial for $G$ centered at $(-2,1)$ .
(b): Is the point $(-2,1,G(-2,1))$ a local min, a local max, a saddle, or none of these?

Consider the differentiable function $H:\R ^2\to \R$ where: $\begin{align*} H(2,2) &= 3 & H^{(1,0)}(2,2) &= 0 & H^{(2,0)}(2,2) &= 3\\ H^{(1,1)}(2,2) &=7 & H^{(0,1)}(2,2) &= 0 & H^{(0,2)}(2,2) &= 2 \end{align*}$

(a): Write down the second degree Taylor polynomial for $H$ centered at $(2,2)$ .
(b): Is the point $(2,2,H(2,2))$ a local min, a local max, a saddle, or none of these?

Consider the differentiable function $I:\R ^2\to \R$ where: $\begin{align*} I(1,-1) &= -4 & I^{(1,0)}(1,-1) &= 1 & I^{(2,0)}(1,-1) &= 2\\ I^{(1,1)}(1,-1) &=-6 & I^{(0,1)}(1,-1) &= 2 & I^{(0,2)}(1,-1) &= 2 \end{align*}$

(a): Write down the second degree Taylor polynomial for $I$ centered at $(1,-1)$ .
(b): Let $\vecl (t)$ parameterize a line from $(0,0)$ to $(2,-2)$ as $t$ runs from $0$ to $1$ . Compute $\eval {\dd {t} F (\vecl (t))}_{t=1/2}$ .
(c): Is the point $(1,-1,I(1,-1))$ a local min, a local max, a saddle, or none of these?

Consider the differentiable function $J:\R ^2\to \R$ where: $\begin{align*} J(-3,1) &= -4 & J^{(1,0)}(-3,1) &= -2 & J^{(2,0)}(-3,1) &= -4\\ J^{(1,1)}(-3,1) &=3 & J^{(0,1)}(-3,1) &= -1 & J^{(0,2)}(-3,1) &= -3 \end{align*}$

(a): Write down the second degree Taylor polynomial for $J$ centered at $(-3,1)$ .
(b): Let $\vecl (t)$ parameterize a line from $(3,-1)$ to $(-6,2)$ as $t$ runs from $0$ to $1$ . Compute $\eval {\dd {t} F(\vecl (t))}_{t=2/3}$ .
(c): Is the point $(-3,1,J(-3,1))$ a local min, a local max, a saddle, or none of these?