Q8Single correct5 Marks21 Dec 2025

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a differentiable function. Consider the following statements. S1: The tangent plane to the graph of f at $(c, c)$ is parallel to the xy-plane. S2: $(c, c)$ is a critical point of f, i.e., grad f(c, c) = 0. Which of the following is true?

A

Neither of the two statements implies the other.

B

S1 holds if and only if S2 holds.

C

S1 implies S2, but S2 does not imply S1.

D

S2 implies S1, but S1 does not imply S2.