Find the stationary points and use the method illustrated in earlier examples to determine the local extreme values.

$$f(x, y)=2 x+2 y-x^2+y^2+5$$

Use differentials to find the approximate value.

$$\sin \frac{6}{7} \pi \cos \frac{1}{5} \pi$$

Maximize $f(x, y, z)=3 x-2 y+z$ on the sphere $x^2+$ $y^2+z^2=14$.

Find $du/dt$ by applying (16.3.5) or (16.3.6).

$$u=e^x \sin y+e^y \sin x ; \quad x=\frac{1}{2} t, y=2 t$$

Find the unit vector in the direction in which $f$ increases most rapidly at $P$ and give the rate of the change of $f$ in that direction.

$$f(x, y)=e^x \arctan y ; \quad P(0,1)$$

Find the stationary points and the local extreme values.

$$f(x, y)=\frac{-2 x}{x^2+y^2+1}$$

Find positive numbers $x, y, z$ such that $x+y+z=18$ and $x y z$ is a maximum. HINT: Maximize $f(x, y)=$ $x y(18-x-y)$ on the triangular region bounded by the positive $x$ - and $y$-axes and the line $x+y=18$.

Find the point(s) on the surface at which the tangent plane is horizontal.

$$x y+a^3 x^{-1}+b^3 y^{-1}-z=0$$

Find all functions with this gradient.

$$\left(x^2 \arcsin y\right) \mathbf{i}+\left(\frac{x^3}{3 \sqrt{1-y^2}}-\ln y\right) \mathbf{j}$$

Find the gradient at $P$.

$$f(x, y)=\ln \left(x^2+y^2\right) ; \quad P(2,1)$$

Find the directional derivative of $f(x, y, z)=x e^{y^2-z^2}$ at $(1,2,-2)$ in the direction of increasing $t$ along the path

$$\mathbf{r}(t)=t \mathbf{i}+2 \cos (t-1) \mathbf{j}-2 e^{t-1} \mathbf{k} .$$

Let $x, y, z$ be the angles of a triangle. Determine the maximum value of $f(x, y, z)=\sin x \sin y \sin z$.

Find the stationary points and the local extreme values.

$$f(x, y)=(x-y)(x y-1)$$

Find the absolute extreme values taken on by $f$ on the set indicated. $f(x, y)=\frac{1}{\sqrt{x^2+y^2}} ; \quad 1 \leq x \leq 3,1 \leq y \leq 4$.