GATE DS&AI 2024 | Question: 5

Question

For any twice differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$, if at some $x^{*} \in \mathbb{R}, f^{\prime}\left(x^{*}\right)=0$ and $f^{\prime \prime}\left(x^{*}\right)>0$, then the function $f$ necessarily has a $\_\_\_\_\_\_\_$ at $x=x^{*}$.

Note: $\mathbb{R}$ denotes the set of real numbers.

• A. local minimum
• B. global minimum
• C. local maximum
• D. global maximum

Answer 1

The given statement describes the second derivative test, which is used to analyze the behavior of a function around a critical point \( x^* \).

If \( f'(x^*) = 0 \) and \( f''(x^*) > 0 \), then \( x^* \) is a local minimum of the function \( f \).

This means that around the point \( x^* \), the function \( f \) decreases to the left of \( x^* \) and increases to the right of \( x^* \), indicating a valley or a bottom point in the graph of \( f \).

Therefore, the correct answer is:

A. local minimum.