91Ó°ÊÓ

[T] By investing \(x\) units of labor and \(y\) units of capital, a watch manufacturer can produce \(P(x, y)=50 x^{0.4} y^{0.6}\) watches. Find the maximum number of watches that can be produced on a budget of \(\$ 20,000\) if labor costs \(\$ 100 /\) unit and capital costs \(\$ 200 /\) unit. Use a CAS to sketch a contour plot of the function.

For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample. The domain of \(f(x, y)=x^{3} \sin ^{-1}(y)\) is \(x=\) all real numbers, and \(-\pi \leq y \leq \pi\)

For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample. If the function \(f(x, y)\) is continuous everywhere, then \(f_{x y}=f_{y x}\)

The linear approximation to the function of \(f(x, y)=5 x^{2}+x \tan (y) \quad\) at \(\quad(2, \pi) \quad\) is given by \(L(x, y)=22+21(x-2)+(y-\pi)\)

\(\left(\frac{3}{4}, \frac{9}{16}\right) \quad\) is a critical point of \(g(x, y)=4 x^{3}-2 x^{2} y+y^{2}-2\)

For the following exercises, sketch the function in one graph and, in a second, sketch several level curves. \(f(x, y)=e^{-\left(x^{2}+2 y^{2}\right)}\)

For the following exercises, sketch the function in one graph and, in a second, sketch several level curves. \(f(x, y)=x+4 y^{2}\)

For the following exercises, evaluate the following limits, if they exist. If they do not exist, prove it. $$\lim _{(x, y) \rightarrow(1,1)} \frac{4 x y}{x-2 y^{2}}$$

For the following exercises, evaluate the following limits, if they exist. If they do not exist, prove it. $$\lim _{(x, y) \rightarrow(0,0)} \frac{4 x y}{x-2 y^{2}}$$

For the following exercises, find the largest interval of continuity for the function. \(f(x, y)=x^{3} \sin ^{-1}(y)\)