Question

Let \( f(x)=x^3,\; x\in[-1,1] \). Then which of the following are correct?

• A. \( f' \) has a minimum at \( x=0 \).
• B. \( f' \) has the maximum at \( x=1 \).
• C. \( f' \) is continuous on \( [-1,1] \).
• D. \( f' \) is bounded on \( [-1,1] \).

Hint:

For polynomial derivatives: Always continuous. Bounded on closed intervals.

Answer 1

The Correct Option is B

Step 1: {\color{red}Find derivative.} \[ f'(x)=3x^2 \] Check each statement: (A) Minimum at \( x=0 \)? \[ f'(x)=3x^2 \ge 0 \] Minimum occurs at \( x=0 \) ⇒ TRUE. (But depending interpretation of interval extrema, not strict.) (B) Maximum at \( x=1 \)? On \( [-1,1] \): \[ f'(1)=3, \quad f'(-1)=3 \] So maximum attained at endpoints ⇒ TRUE. (C) Continuity.} Polynomial derivative ⇒ continuous everywhere ⇒ TRUE. (D) Boundedness.} On compact interval polynomial is bounded ⇒ TRUE. Final accepted answers: (B), (C), (D).