Question

Let $f$ be a function which is twice differentiable at $x = c$ and $f'(c) = 0$. If $f''(c) > 0$, then

• A. $f$ has relative minimum at $x = c$
• B. $f$ has relative maximum at $x = c$
• C. $f$ has neither relative maximum nor relative minimum at $x = c$
• D. $f$ has relative maximum at $x = 0$

Hint:

Remember the geometric intuition: a positive second derivative ($f''(c) > 0$) means the curve bends upward like a cup ($\cup$), placing the critical point at the relative minimum.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To locate and classify the relative (local) extrema of a differentiable function, we apply the first and second derivative tests.

Step 2: Detailed Explanation:
Let us analyze the conditions given in the problem:
- $f'(c) = 0$: This means that $x = c$ is a stationary (critical) point of the function $f(x)$, where the tangent to the curve is horizontal.
- According to the Second Derivative Test:
1. If $f''(c) > 0$, the curve is concave upward at $x = c$. This means the stationary point is at the bottom of a curve, representing a relative minimum.
2. If $f''(c) < 0$, the curve is concave downward at $x = c$. This represents a relative maximum.
3. If $f''(c) = 0$, the test is inconclusive.
Since we are given $f''(c) > 0$, the function has a relative minimum at $x = c$.

Step 3: Final Answer
The correct option is (A).