91Ó°ÊÓ

Find all the local maxima, local minima, and saddle points of the functions. $$f(x, y)=x^{2}+2 x y$$

Use the method of Lagrange multipliers to find the dimensions of the rectangle of greatest area that can be inscribed in the ellipse \(x^{2} / 16+y^{2} / 9=1\) with sides parallel to the coordinate axes.

Find all the local maxima, local minima, and saddle points of the functions. $$f(x, y)=\sqrt{56 x^{2}-8 y^{2}-16 x-31}+1-8 x$$

Find the limits. $$\lim _{(x, y) \rightarrow(1, \pi / 6)} \frac{x \sin y}{x^{2}+1}$$

(a) express and \(2 x,\) aut \(\partial y,\) and aut \(/\) az as functions of \(x, y,\) and \(z\) both by using the Chain Rule and by expressing \(u\) directly in terms of \(x, y,\) and \(z\) before differentiating. Then (b) evaluate \(\partial u / \partial x, \partial u / \partial y,\) and and \(| \partial z\) at the given point \((x, y, z)\) $$\begin{aligned} &u=\frac{p-q}{q-r}, p=x+y+z, q=x-y+z\\\ &r=x+y-z ; \quad(x, y, z)=(\sqrt{3}, 2,1) \end{aligned}$$

In Exercises find \(\partial f / \partial x\) and \(\partial f / \partial y\). $$f(x, y)=(x+y) /(x y-1)$$

Find an equation for the plane that is tangent to the given surface at the given point. $$z=\sqrt{y-x}, \quad(1,2,1)$$

Find and sketch the domain for each function. $$f(x, y)=\sqrt{\left(x^{2}-4\right)\left(y^{2}-9\right)}$$

Find the dimensions of the rectangle of largest perimeter that can be inscribed in the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\) with sides parallel to the coordinate axes. What is the largest perimeter?

Find the limits. $$\lim _{(x, y) \rightarrow(\pi / 2,0)} \frac{\cos y+1}{y-\sin x}$$