Find the critical points of each of the functions and classify them as local maxima, minima, or neither. $f(x, y)=x^2-y^2+x y-7$.

Let $f(x, y)=x^2+x y+y^2$. (a) Find the maximum and minimum points and values of $f$ along the circle $x^2+y^2=1$. (b) Moving counterclockwise along the circle $x^2+y^2=1$, is the function increasing or decreasing at the points $(\pm 1,0)$ and $(0, \pm 1)$ ? (c) Find extreme points and values for $f$ in the disk $D$ consisting of all $(x, y)$ such that $x^2+y^2 \leqslant 1$.

Suppose that $x$ and $y$ are related by the equations given in Exercise. Find $d y / d x$ at the indicated points. $\int_x^y u^2 d u=5 \frac{1}{3}, x=-2, y=2$.

Suppose that $x$ and $y$ are related by the equations given in Exercise. Find $d y / d x$ at the indicated points. $x^4+y^4=17, x=-1, y=2$.

Verify the chain rule for the functions and curves. $f(x, y, z)=x y+y z+x z ; \sigma(t)=(t, t, t)$.

Find the critical points of each of the functions and classify them as local maxima, minima, or neither. $f(x, y)=x^2+y^2+3 x-2 y+1$.

A firm uses wool and cotton fiber to produce cloth. The amount of cloth produced is given by $Q(x, y)=x y-x-y+1$, where $x$ is the number of pounds of wool, $y$ is the number of pounds of cotton, and $x>1$ and $y>1$. If wool costs $p$ dollars per pound, cotton costs $q$ dollars per pound, and the firm can spend $B$ dollars on material, what should the mix of cotton and wool be to produce the most cloth?

Find the equation for the tangent line to each curve at the indicated point. $x y=17 ;\left(x_0, 17 / x_0\right)$.

Find the equation for the tangent line to each curve at the indicated point. $x^2+2 y^2=3 ;(1,1)$.

Suppose that $x$ and $y$ are related by the equations given in Exercise. Find $d y / d x$ at the indicated points. $x+\cos y=1, x=1, y=\pi / 2$.

Find the critical points of each of the functions and classify them as local maxima, minima, or neither. $f(x, y)=x^2+y^2+6 x-4 y+13$.

Verify the chain rule for the functions and curves. $f(x, y, z)=\sqrt{x^2+y^2+z^2} ; \sigma(t)=(\sin t, \cos t, t)$.

A water main consists of two sections of pipe of fixed lengths, $l_1, l_2$ carrying fixed amounts $Q_1$ and $Q_2$ liters per second. For a given total loss of head $h$, the (variable) diameters $D_1, D_2$ of the pipe will result in a minimum cost if

$$C=l_1\left(a+b D_1\right)+l_2\left(a+b D_2\right)=\text { minimum }$$

subject to the condition

$$h=\frac{c l_1 Q_1^2}{D_1^5}+\frac{c l_2 Q_2^2}{D_2^5} .$$

Find the ratio $D_2 / D_1$.

Suppose that $x=f(t)$ and $y=g(t)$ satisfy the relation. Relate $d x / d t$ and $d y / d t$. $\tan ^{-1}(x-y)=\pi / 4$