List of top Mathematics Questions on Optimization asked in UP Board XII

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.