GATE 2023 BM – Question 53
For f(x)=x⁴−x² on the real line, which statements are correct?
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Show answer and explanation
Correct answer: (A) f(x) is even; (C) f(x) has two minima
Explanation
Only even powers occur, so f(−x)=f(x), A. Differentiating gives f′=2x(2x²−1), with stationary points zero and ±1/√2. The latter have positive second derivative and value 1/4−1/2=−1/4; hence there are two minima, C, but their value is not −.5. It is not odd. Thus A,C.