Determine whether the statement is true or folse. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

If $f(x, y)$ has two local maxima, then $f$ must have a local minimum.

A metal plate is situated in the $x y$-plane and occupies the rectangle $0 \leqslant x \leqslant 10,0 \leqslant y \leqslant 8$, where $x$ and $y$ are measured in meters. The temperature at the point $(x, y)$ in the plate is $T(x, y)$, where $T$ is measured in degrees Celsius. Temperatures at equally spaced points were measured and recorded in the table.

Find a linear approximation to the temperature function $T(x, y)$ near the point $(6,4)$. Then use it to estimate the temperature at the point $(5,3.8)$.

Suppose $f$ is a differentiable function of $x$ and $y$, and $g(r, s)=f\left(2 r-s, s^2-4 r\right)$. Use the table of values in Exercise 11 to calculate $g_r(1,2)$ and $g_s(1,2)$.

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function.

$$f(x, y)=x^2+y^2+\frac{1}{x^2 y^2}$$

A metal plate is situated in the $x y$-plane and occupies the rectangle $0 \leqslant x \leqslant 10,0 \leqslant y \leqslant 8$, where $x$ and $y$ are measured in meters. The temperature at the point $(x, y)$ in the plate is $T(x, y)$, where $T$ is measured in degrees Celsius. Temperatures at equally spaced points were measured and recorded in the table.

Estimate the value of $T_{x y}(6,4)$.

A metal plate is situated in the $x y$-plane and occupies the rectangle $0 \leqslant x \leqslant 10,0 \leqslant y \leqslant 8$, where $x$ and $y$ are measured in meters. The temperature at the point $(x, y)$ in the plate is $T(x, y)$, where $T$ is measured in degrees Celsius. Temperatures at equally spaced points were measured and recorded in the table.

Estimate the value of $D_{\mathbf{u}} T(6,4)$, where $\mathbf{u}=(\mathbf{i}+\mathbf{j}) / \sqrt{2}$. Interpret your result.

A metal plate is situated in the $x y$-plane and occupies the rectangle $0 \leqslant x \leqslant 10,0 \leqslant y \leqslant 8$, where $x$ and $y$ are measured in meters. The temperature at the point $(x, y)$ in the plate is $T(x, y)$, where $T$ is measured in degrees Celsius. Temperatures at equally spaced points were measured and recorded in the table.

Estimate the values of the partial derivatives $T_x(6,4)$ and $T_y(6,4)$. What are the units?

State the Chain Rule for the case where $z=f(x, y)$ and $x$ and $y$ are functions of one variable. What if $x$ and $y$ are functions of two variables?

Determine whether the statement is true or folse. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

If $f(x, y)=\sin x+\sin y$, then $-\sqrt{2} \leqslant D_{\text {u }} f(x, y) \leqslant \sqrt{2}$.

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function.

$$f(x, y)=e^x \cos y$$

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).

$$f(x, y, z)=x^2+y^2+z^2 ; \quad x^4+y^4+z^4=1$$

Suppose $f$ is a differentiable function of $x$ and $y$, and $g(u, v)=f\left(e^u+\sin v, e^u+\cos v\right)$. Use the table of values to calculate $g_\nu(0,0)$ and $g_\nu(0,0)$.

Find the limit, if it exists, or show that the limit does not exist.

$$\lim\limits_{(x, y) \rightarrow(0,0)} \frac{x y}{\sqrt{x^2+y^2}}$$

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).

$$f(x, y, z)=x^2 y^2 z^2 ; \quad x^2+y^2+z^2=1$$