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 $f(\sigma(t))$ is an increasing function of $t$. What can you say about the angle between the gradient $\nabla f$ and the velocity vector $\sigma^{\prime}$ ?

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. $f(x, y)=x^2+y^2+3 x-2 y+1$.

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

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?

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

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. $f(x, y)=x^2+y^2+6 x-4 y+13$.

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 $D_2 / D_1$.

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

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

Verify the chain rule for the functions and curves. $f(x, y, z)=e^{x y z} ; \sigma(t)=\left(6 t, 3 t^2, t^3\right)$.

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