Question

If \( f(x) = \frac{x^2}{2}, \text{for} x \leq 0, \frac{2\sin x}{x}, \text{for} x>0 \), then \( x = 0 \) is:

• A. point of minima
• B. point of maxima
• C. point of discontinuity
• D. None of the above

Hint:

Check the derivatives and limits of piecewise functions to classify critical points.

Answer 1

The Correct Option is A

Step 1: Analyze the piecewise function.
The function is continuous at \( x = 0 \), but to determine if it is a point of minima or maxima, we compute its derivative and check for concavity.
Step 2: Conclusion.
Thus, \( x = 0 \) is a point of minima.
Final Answer: \[ \boxed{\text{point of minima}} \]