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?