Answer
Minimum height=$1$ at $ (±1,0)$
Maximum height =$4 $ at $(0,±1)$
Work Step by Step
\[
x^{2}+4 y^{2}=z
\]
The constraint is $g(x, y,)=0=x^{2}+y^{2}-1$
We find the least amount for its construction $f$ by using Lagrange multipliers,
\[
\lambda \nabla g=\nabla z
\]
\[
\begin{aligned}
\nabla z &=2 x \hat{i}+8 y \hat{j} \\
\nabla g &=2 x \hat{i}+2 y \hat{j} \\
\nabla z &=\lambda \nabla g \\
2 x \hat{i}+8 y \hat{j} &=\lambda 2 x \hat{i}+2 y \hat{j}
\end{aligned}
\]
\[
\begin{array}{l}
2 x \lambda=2x \\
\lambda(2 y)=8y
\end{array}
\]
$y \neq 0,$ and then $4=\lambda$
\[
0=x
\]
\[
\begin{aligned}
1=y^{2} & \\
4 &=x^{2}+4 y^{2}=z
\end{aligned}
\]
$0=y,$ and then
\[
\begin{aligned}
1=x^{2} & \\
1=z &
\end{aligned}
\]
Maximum height at (0,±1)
\[
\begin{array}{l}
z=x^{2}+4 y^{2} \\
z=0+4(1) \\
4=z
\end{array}
\]
Minimum height at (±1,0)
\[
\begin{array}{l}
z=x^{2}+4 y^{2} \\
1=z
\end{array}
\]
