Q23Multiple correct3 Marks13 Apr 2025

Let $f: \mathbb{R}^n \to \mathbb{R}$ be a scalar-valued multivariable function ($n \ge 2$) such that the partial derivatives exist at all points and are continuous. Choose the correct statements from the following.

A

For $x \in \mathbb{R}^n$, if the directional derivatives at $x$ in all directions are zero, then $x$ is a critical point of $f$.

B

The second order partial derivatives satisfy the relation $\frac{\partial^2 f}{\partial x_i \partial x_j} = \frac{\partial^2 f}{\partial x_j \partial x_i}$ for all $i, j$, whenever they exist.

C

The i^th component of $\nabla$ f gives the partial derivative of f with respect to the i^th variable v_i.

D

For each point, there exists a unit direction u such that the directional derivative of f at that point in the direction u is zero.