List of top Mathematics Questions on Optimization

Divide a number 15 into two parts such that the square of one multiplied with the cube of the other is a maximum.
A wire of length 28m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and circle is minimized?
A box with a square base is to have an open top. The surface area of the box is 147 cm\(^2\). What should its dimensions be in order that the volume is largest?

Prove that the height of the cylinder of maximum volume inscribed in a sphere of radius \( R \) is \( \frac{2R}{\sqrt{3}} \). 

(e) Show that the semi-vertical angle of the right circular cone of maximum volume and given slant height is \( \tan^{-1} (\sqrt{2}) \):
(b) Minimize \( Z = x + 2y \) under the following constraints: \[ 2x + y \geq 3, \quad x + 2y \geq 6, \quad x \geq 0, \quad y \geq 0. \]
Find two positive numbers whose sum is 15 and the sum of their squares is minimum.
An Apache helicopter of the enemy is flying along the curve \[ y = x^2 + 7. \] A soldier placed at the point \( (3, 7) \), wants to shoot down the helicopter when it is nearest to him. Find the nearest distance.
The maximum value of the function \[ y = \frac{x^2}{x^4 + 4}, x \in \mathbb{R}, \] is ............

Consider the domain \( D = \{ (x, y) \in \mathbb{R}^2 : x \leq y \} \) and the function \( h : D \to \mathbb{R} \) defined by \[ h((x, y)) = (x - 2)^4 + (y - 1)^4, (x, y) \in D. \] Then the minimum value of \( h \) on \( D \) equals 
 

Suppose \( x, y, z \) are positive real numbers such that \( x + 2y + 3z = 1 \). If \( M \) is the maximum value of \( xyz^2 \), then the value of \( \frac{1}{M} \) is ..................
Let \( P \) be the point on the surface \[ z = \sqrt{x^2 + y^2} \] closest to the point \( (4,2,0) \). Then the square of the distance between the origin and \( P \) is ................
Consider the function \( f(x, y) = 5 - 4 \sin x + y^2 \) for \( 0<x<2\pi \) and \( y \in \mathbb{R} \). The set of critical points of \( f(x, y) \) consists of