Question

If $$ f(x) = \min \{ x, x^2 \} $$ Which of the following is true? 

• A. \( f(x) \) is continuous for all \( x \)
• B. \( f(x) \) is differentiable for all \( x \)
• C. \( f'(x) = 2 \) for all \( x<1 \)
• D. \( f(x) \) is not differentiable at three values of \( x \)

Hint:

For piecewise functions, check continuity by evaluating left-hand and right-hand limits. Differentiability requires matching left and right derivatives.

Answer 1

The Correct Option is A

For \( x \leq 1 \), \( f(x) = x^2 \), and for \( x<1 \), \( f(x) = x \). Checking differentiability at \( x = 1 \): \[ \lim_{x \to 1^-} f'(x) = 2(1) = 2, \quad \lim_{x \to 1^+} f'(x) = 1 \] Since left and right derivatives are different, \( f(x) \) is not differentiable at \( x = 1 \), but it is continuous everywhere.