Question:
Let \( f(x)=x^3,\; x\in[-1,1] \). Then which of the following are correct?
Let \( f(x)=x^3,\; x\in[-1,1] \). Then which of the following are correct?
Show Hint
For polynomial derivatives:
Always continuous.
Bounded on closed intervals.
Updated On: Mar 3, 2026
- \( f' \) has a minimum at \( x=0 \).
- \( f' \) has the maximum at \( x=1 \).
- \( f' \) is continuous on \( [-1,1] \).
- \( f' \) is bounded on \( [-1,1] \).
Show Solution
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The Correct Option is B
Solution and Explanation
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).
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