Problems on Maxima and Minima of Functions

Prove: if the function $f(x)$ is nonnegative, then the function

has the same extremum points as the function $f(x)$.

Prove: if, for $-\infty <x<+\infty$, the function $\varphi (x)$ is monotonically increasing, then the functions

have the same extremum points.

The sum of two positive numbers equals the constant $a$. Find the maximum value of the product of the $m$-th power and $n$-th power ($m>0, n>0$) of these two positive numbers.

The product of two positive numbers equals the constant $a$. Find the minimum value of the sum of the $m$-th power and $n$-th power ($m>0, n>0$) of these two numbers.

For what base should logarithms be taken so that there is a number equal to its own logarithm?

Among all rectangles of area $S$, find the one with the smallest perimeter.

If the sum of one leg of a right triangle and the hypotenuse is a constant, find the right triangle with the maximum area.

What dimensions give a closed cylindrical can of volume $V$ the minimum surface area?

Inscribe a rectangle of maximum area in a given circular segment not exceeding a semicircle.

Inscribe, in the ellipse

a rectangle of maximum area with sides parallel to the axes of the ellipse.

Inscribe, in a triangle with base $b$ and height $h$, a rectangle of maximum perimeter, and investigate the possibility of a solution to this problem.

A beam with a rectangular cross-section is cut from a circular log of diameter $d$; the base of this rectangle equals $b$, and the height equals $h$. If the strength of the beam is proportional to $bh^{2}$, what dimensions of the beam give the maximum strength?

Inscribe, in a hemisphere of radius $R$, a right rectangular parallelepiped with a square base and of maximum volume.

Inscribe, in a sphere of radius $R$, a cylinder of maximum volume.

Inscribe, in a sphere of radius $R$, a cylinder of maximum surface area.

Construct, about a given sphere, a circumscribed cone of minimum volume.

Find the maximum volume of a cone with slant height $l$.

Inscribe, in a right circular cone with vertex angle $2\alpha$ and base radius $R$, a cylinder of maximum surface area.

Find the shortest distance from the point $M(p,p)$ to the parabola $y^{2}=2px$.

Find the shortest and longest distances from the point $A(2,0)$ to the circle $x^{2}+y^{2}=1$.

Find the longest chord of the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ ($0<b<a$) passing through the vertex $(0,-b)$.

Draw a tangent line at the point $M(x,y)$ on the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$; this tangent line, together with the coordinate axes, forms a triangle. Find the point $M$ for which the area of this triangle is minimum.

A body has the shape of a right circular cylinder with a hemispherical top. If the volume of this body equals $V$, what dimensions of the body give the minimum surface area?

The cross-section of an open water channel is an isosceles trapezoid. If the cross-sectional area of the flowing water equals $S$, and the height of the water surface equals $h$, what should the inclination angle $\varphi$ of the channel's sides be so that the wetted perimeter of the cross-section is minimum?

Let the area enclosed by a closed curve be $S$, and let a circle also enclose the same area $S$; then the ratio of the length of the closed curve to the circumference of the circle is called the "curvature" of the curve.

Let the isosceles trapezoid $ABCD$ ($AD \parallel BC$) have base $AD=2a$ and acute angle $BAD=\alpha$. What shape should the isosceles trapezoid have for the minimum curvature?

What sector must be cut away from a circle of radius $R$ so that the remaining part, rolled up, forms a funnel of maximum volume?

A railway running from south to north passes through town $B$. A factory $A$ is at a shortest distance $a$ kilometers from this railway, and $b$ kilometers from town $B$ to the north. In order to transport goods from $A$ to $B$ most economically, a spur track is to be built from the factory. If the cost of transporting one ton of goods along the spur is $p$ rubles per kilometer, and along the railway is $q$ rubles per kilometer ($p>q$), at what angle $\varphi$ to the railway should the spur be laid?

Two ships travel in straight lines with constant speeds $u$ and $v$ respectively, the angle between their two directions of travel being $\theta$. If at a certain moment their distances from the point where their paths cross are $a$ and $b$ respectively, find the minimum distance between the two ships.

There are light sources at points $A$ and $B$, with intensities $S_{1}$ candlepower and $S_{2}$ candlepower respectively. Find the point $M$ of minimum illumination on the segment $AB=a$.

A luminous point lies on the line joining the centers of two non-intersecting spheres of radii $R$ and $r$ ($R>r$), outside both spheres. Where should this luminous point be located so that the sum of the illuminated portions of the surfaces of the two spheres is maximal?

Let the radius of a round table top be $a$. At what height above the center of the table top should a lamp be placed so that the illuminance at the edge of the table is maximal?

A canal $b$ meters wide is to be built connecting to a river $a$ meters wide, meeting it at a right angle. What is the maximum length of a boat that can enter this canal?

The cost of a ship's voyage over a day and night consists of two parts: a fixed part equal to $a$ rubles, and a variable part that increases in proportion to the cube of the speed. At what speed $v$ is the voyage most economical?

A body of weight $P$ lies on a rough horizontal surface and must be moved from its position by an applied force. If the coefficient of friction of the body is $k$, at what angle of inclination of the applied force to the horizontal surface is the required force minimal?

A teacup has the shape of a hemisphere of radius $a$. A rod of length $l>2a$ is placed in the cup. Find the equilibrium position of the rod.