91Ó°ÊÓ

Use Lagrange multipliers to solve the given optimization problem. HINT [See Example 2.] Find the minimum value of \(f(x, y)=x^{2}+y^{2}\) subject to \(x y^{2}=16\). Also find the corresponding point(s) \((x, y)\).

Calculate \(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y},\left.\frac{\partial f}{\partial x}\right|_{(1,-1)}\), and \(\left.\frac{\partial f}{\partial y}\right|_{(1,-1)}\) when defined. HINT [See Quick Examples page 1098.] $$ f(x, y)=\frac{1}{(x y+1)^{2}} $$

Compute the integrals. HINT [See Example 1.] $$ \int_{0}^{1} \int_{0}^{x} e^{x^{2}} d y d x $$

Locate and classify all the critical points of the functions. HINT [See Example 2.] $$ g(x, y)=1-x^{2}-x-y^{2}+y $$

Calculate \(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y},\left.\frac{\partial f}{\partial x}\right|_{(1,-1)}\), and \(\left.\frac{\partial f}{\partial y}\right|_{(1,-1)}\) when defined. HINT [See Quick Examples page 1098.] $$ f(x, y)=e^{x+y} $$

Classify each function as linear, interaction, or neither. HINT [See Quick Examples page 1083.] $$ P\left(x_{1}, x_{2}, x_{3}\right)=0.4+2 x_{1}-x_{3} $$

Calculate \(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y},\left.\frac{\partial f}{\partial x}\right|_{(1,-1)}\), and \(\left.\frac{\partial f}{\partial y}\right|_{(1,-1)}\) when defined. HINT [See Quick Examples page 1098.] $$ f(x, y)=e^{2 x+y} $$

Locate and classify all the critical points of the functions. HINT [See Example 2.] $$ g(x, y)=x^{2}+x+y^{2}-y-1 $$

Compute the integrals. HINT [See Example 1.] $$ \int_{0}^{1} \int_{0}^{x^{2}} e^{x^{3}+1} d y d x $$

Calculate \(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y},\left.\frac{\partial f}{\partial x}\right|_{(1,-1)}\), and \(\left.\frac{\partial f}{\partial y}\right|_{(1,-1)}\) when defined. HINT [See Quick Examples page 1098.] $$ f(x, y)=5 x^{0.6} y^{0.4} $$