2.9 Maximum and Minimum Values

Find the high and low points of the surface $z=\sqrt{x^2+y^2}$ with $(x,y)$ varying over the square $|x|\le 1$, $|y|\le 1$. Discuss the values of $z_x,\ z_y$ there. Do not evaluate any derivatives in answering this question.

If $t_0$ is a local minimum or maximum of the smooth function $f(t)$ of one variable ($t$ runs over all real numbers) then $f'(t_0)=0.$ Derive an analogous necessary condition for $\vx_0$ to be a local minimum or maximium of the smooth function $g(\vx)$ restricted to points on the line $\vx=\va+t\vd\ .$ The test should involve the gradient of $g(\vx)$.

Let $z = f(x,y) = {(y^2 - x^2)}^2$.

Use the Second Derivative Test to find all values of the constant $c$ for which the function $z = x^2 + cxy + y^2$ has a saddle point at $(0,0)$.

Find and classify all critical points of the function

Find all critical points for $f(x,y) = x(x^2 + xy + y^2 - 9)$. Also find out which of these points give local maximum values for $f(x,y)$, which give local minimum values, and which give saddle points.

Find the largest and smallest values of $x^2 y^2 z$ in the part of the plane $2x + y + z = 5$ where $x \ge 0$, $y \ge 0$ and $z \ge 0$. Also find all points where those extreme values occur.

Find and classify all the critical points of $f(x,y)=x^2+y^2+x^2y+4$.

Find all saddle points, local minima and local maxima of the function

For the surface

Find and classify [as local maxima, local minima, or saddle points] all critical points of $f(x,y)$.

Find the maximum and minimum values of $f(x,y)=xy-x^3y^2$ when $(x,y)$ runs over the square $0\le x\le 1$, $0\le y\le 1$.

The temperature at all points in the disc $x^2+y^2\le 1$ is given by $T(x,y)=(x+y)e^{-x^2-y^2}$. Find the maximum and minimum temperatures at points of the disc.

The images below depict level sets $f (x, y) = c$ of the functions in the list at heights $c = 0, 0.1, 0.2, \ldots , 1.9, 2$. Label the pictures with the corresponding function and mark the critical points in each picture. (Note that in some cases, the critical points might not be drawn on the images already. In those cases you should add them to the picture.)

Let the function

Classify as $\big[$local maxima, minima or saddle points$\big]$ all critical points of $f(x,y)$.

Let $h(x, y) = y(4 - x^2 - y^2)$.

Find the absolute maximum and minimum values of the function $f(x, y) = 5 + 2x - x^2 - 4y^2$ on the rectangular region

Find the minimum of the function $h(x,y) = -4x - 2y + 6$ on the closed bounded domain defined by $x^2 + y^2 \le 1$.

Let $f(x,y) = xy(x + y - 3)$.

Find and classify the critical points of $f(x,y) = 3x^2 y + y^3 - 3x^2 - 3y^2 + 4$.

Consider the function

Find all critical points of the function $f(x,y)=x^4+y^4-4xy+2$, and for each determine whether it is a local minimum, maximum or saddle point.

Let

Find all the critical points of the function

defined in the $xy$-plane. Classify each critical point as a local minimum, maximum or saddle point.

A metal plate is in the form of a semi-circular disc bounded by the $x$-axis and the upper half of $x^2+y^2=4$. The temperature at the point $(x,y)$ is given by $T(x,y)=\ln\big(1+x^2+y^2\big)-y$. Find the coldest point on the plate, explaining your steps carefully. (Note: $\ln 2\approx 0.693$, $\ln 5\approx 1.609$)

Find all the critical points of the function

defined in the $xy$-plane. Classify each critical point as a local minimum, maximum or saddle point. Explain your reasoning.

Consider the function $g(x,y)=x^2-10y-y^2 .$

Find and classify all critical points of

Find the maximum value of

on the quarter-circle $D = \Set{(x, y)} { x^2 + y^2 \le 4,\ x\ge0,\ y\ge 0}$.

Equal angle bends are made at equal distances from the two ends of a 100 metre long fence, so that the resulting three segment fence can be placed along an existing wall to make an enclosure of trapezoidal shape. What is the largest possible area for such an enclosure?

Find the most economical shape of a rectangular box that has a fixed volume $V$ and that has no top.

The temperature $T(x,y)$ at a point of the $xy$–plane is given by

Consider the function

where $k > 0$ is a constant. Find and classify all critical points of $f(x,y)$ as local minima, local maxima, saddle points or points of indeterminate type. Carefully distinguish the cases $k < \frac{1}{2}$, $k = \frac{1}{2}$ and $k > \frac{1}{2}$.

An experiment yields data points $(x_i,y_i),\ i=1,2,\cdots,n.$ We wish to find the straight line $y=mx+b$ which “best" fits the data. The definition of “best" is “minimizes the root mean square error", i.e. minimizes $\sum_{i=1}^n (mx_i+b-y_i)^2$. Find $m$ and $b$.