91Ó°ÊÓ

Sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced. $$f(x, y)=-x-y$$

Find all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these. $$f(x, y)=y^{3}-3 x y+6 x$$

Find the partial derivatives in Problems. The variables are restricted to a domain on which the function is defined. $$f_{x} \text { and } f_{y} \text { if } f(x, y)=10 x^{2} e^{3 y}$$

Find all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these. $$f(x, y)=x^{3}+y^{2}-3 x^{2}+10 y+6$$

Sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced. $$f(x, y)=y-x^{2}$$

The monthly mortgage payment in dollars, \(P\), for a house is a function of three variables: $$ P=f(\boldsymbol{A}, \boldsymbol{r}, \boldsymbol{N}) $$ where \(A\) is the amount borrowed in dollars, \(r\) is the interest rate, and \(N\) is the number of years before the mortgage is paid off. (a) \(f(92000,14,30)=1090.08 .\) What does this tell you, in financial terms? (b) \(\left.\frac{\partial P}{\partial r}\right|_{(92000,14,30)}=72.82 .\) What is the financial significance of the number \(72.82 ?\) (c) Would you expect \(\partial P / \partial A\) to be positive or negative? Why? (d) Would you expect \(\partial P / \partial N\) to be positive or negative? Why?

Find the partial derivatives in Problems. The variables are restricted to a domain on which the function is defined. $$z_{x} \text { if } z=x^{2} y+2 x^{5} y$$

Use Lagrange multipliers to find the maximum or minimum values of \(f(x, y)\) subject to the constraint. $$f(x, y)=x^{2}+y^{2}, \quad x^{4}+y^{4}=2$$

Find the partial derivatives in Problems. The variables are restricted to a domain on which the function is defined. $$\frac{\partial}{\partial m}\left(\frac{1}{2} m v^{2}\right)$$

Find all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these. $$f(x, y)=x^{3}+y^{3}-6 y^{2}-3 x+9$$