Question

For the function \( f(x) = x^3 \), \( x = 0 \) is a point of:

• (A) local maxima
• (B) local minima
• (C) non-differentiability
• (D) inflexion

Answer 1

The Correct Option is D

Step 1: Understand the function
The given function is f(x) = x³. This is a cubic function that is continuous and differentiable everywhere.

Step 2: Find the first derivative
Calculate the first derivative to check for stationary points:
f'(x) = 3x²
At x = 0, f'(0) = 0, so x = 0 is a critical point.

Step 3: Find the second derivative
Calculate the second derivative to determine the nature of the critical point:
f''(x) = 6x
At x = 0, f''(0) = 0, so the second derivative test is inconclusive.

Step 4: Analyze the change of concavity
Check the sign of f''(x) around 0:
- For x < 0, f''(x) < 0 (concave down)
- For x > 0, f''(x) > 0 (concave up)
The concavity changes from concave down to concave up as x passes through zero.

Step 5: Conclusion
Since the concavity changes at x = 0, the point is an inflection point, where the curve changes curvature but is neither a local maxima nor minima.

Final Answer: (D) inflexion