List of top Mathematics Questions on Optimization asked in IIT JAM MA

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