Question

The function f(x) - x³ , x ∈ R has :

• A. Maximum value at x = 0
• B. Minimum value at x = 0
• C. Neither maximum and nor minimum value at x = 0
• D. Maximum value and minimum value at x = 0

Answer 1

The Correct Option is C

The given function is \( f(x) = x^3 \). To analyze the function for maximum or minimum values, we begin by finding its derivative.

Step 1: First Derivative

\( f'(x) = \frac{d}{dx}(x^3) = 3x^2 \)

Step 2: Find Critical Points

Set the first derivative equal to zero:

\( 3x^2 = 0 \Rightarrow x = 0 \)

Step 3: Second Derivative Test

Compute the second derivative:

\( f''(x) = \frac{d}{dx}(3x^2) = 6x \)

Evaluate at \( x = 0 \):

\( f''(0) = 6 \times 0 = 0 \)

Since \( f''(0) = 0 \), the second derivative test is inconclusive.

Step 4: Analyze Behavior Around \( x = 0 \)

Check the sign of the first derivative on either side of \( x = 0 \):

• For \( x < 0 \), try \( x = -1 \): \( f'(-1) = 3(-1)^2 = 3 > 0 \) ⇒ \( f(x) \) is increasing.
• For \( x > 0 \), try \( x = 1 \): \( f'(1) = 3(1)^2 = 3 > 0 \) ⇒ \( f(x) \) is increasing.

Conclusion: Since the function is increasing on both sides of \( x = 0 \), it is not a local maximum or minimum. Thus,

\[ \boxed{\text{Neither maximum nor minimum at } x = 0} \]