극값과 안장점

For the following exercises, find all critical points.

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

For the following exercises, find all critical points.

$f(x,y)={(3x-2)}^{2}+{(y-4)}^{2}$

For the following exercises, find all critical points.

$f(x,y)={x}^{4}+{y}^{4}-16xy$

For the following exercises, find all critical points.

$f(x,y)=15{x}^{3}-3xy+15{y}^{3}$

For the following exercises, find the critical points of the function by using algebraic techniques (completing the square) or by examining the form of the equation. Verify your results using the partial derivatives test.

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

For the following exercises, find the critical points of the function by using algebraic techniques (completing the square) or by examining the form of the equation. Verify your results using the partial derivatives test.

$f(x,y)=-{x}^{2}-5{y}^{2}+8x-10y-13$

For the following exercises, find the critical points of the function by using algebraic techniques (completing the square) or by examining the form of the equation. Verify your results using the partial derivatives test.

$f(x,y)={x}^{2}+{y}^{2}+2x-6y+6$

For the following exercises, find the critical points of the function by using algebraic techniques (completing the square) or by examining the form of the equation. Verify your results using the partial derivatives test.

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

Find the absolute extrema of the given function on the indicated closed and bounded set $R.$

$f(x,y)=xy-x-3y;$ $R$ is the triangular region with vertices $(0,0),(0,4),\,\text{and}\,(5,0).$

Find the absolute maximum and minimum values of $f(x,y)={x}^{2}+{y}^{2}-2y+1$ on the region $R=\{(x,y)|{x}^{2}+{y}^{2}\le 4\}.$

Find the absolute extrema of the given function on the indicated closed and bounded set $R.$

$f(x,y)={x}^{3}-3xy-{y}^{3}$ on $R=\{(x,y):\,-2\le x\le 2,-2\le y\le 2\}$

Find the absolute extrema of the given function on the indicated closed and bounded set $R.$

$f(x,y)=\frac{-2y}{{x}^{2}+{y}^{2}+1}$ on $R=\{(x,y):\,{x}^{2}+{y}^{2}\le 4\}$

Find three positive numbers the sum of which is $27,$ such that the sum of their squares is as small as possible.

Find the points on the surface ${x}^{2}-yz=5$ that are closest to the origin.

The sum of the length and the girth (perimeter of a cross-section) of a package carried by a delivery service cannot exceed $108$ in. Find the dimensions of the rectangular package of largest volume that can be sent.

Find the point on the surface $f(x,y)={x}^{2}+{y}^{2}+10$ nearest the plane $x+2y-z=0.$ Identify the point on the plane.

Find the point in the plane $2x-y+2z=16$ that is closest to the origin.

A shipping company handles rectangular boxes provided the sum of the length, width, and height of the box does not exceed $96$ in. Find the dimensions of the box that meets this condition and has the largest volume.

In Exercises 7–12, functions $z=f(x,y)$, $x=g(t)$ and $y=h(t)$ are given.

(a) Use the Multivariable Chain Rule to compute $\displaystyle \frac{dz}{dt}$. (b) Evaluate $\displaystyle \frac{dz}{dt}$ at the indicated $t$-value.

$z=3x+4y$, $x=t^2$, $y=2t$; $t=1$

In Exercises 7–12, functions $z=f(x,y)$, $x=g(t)$ and $y=h(t)$ are given.

(a) Use the Multivariable Chain Rule to compute $\displaystyle \frac{dz}{dt}$. (b) Evaluate $\displaystyle \frac{dz}{dt}$ at the indicated $t$-value.

$\displaystyle z=x^2-y^2$, $x=t$, $y=t^2-1$; $t=1$