GATE 2017 EC (EC2) – Question 40
The minimum value of $f(x)=\tfrac13x(x^2-3)$ on $-100\le x\le100$ occurs at x=________.
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Correct answer: -100.01 to -99.99
Explanation
Differentiate: $f'(x)=x^2-1$, so the interior stationary points are x=−1 and x=1. A global minimum on a closed interval can also occur at an endpoint.
The stationary values are $f(-1)=2/3$ and $f(1)=-2/3$. The endpoint values are $f(-100)=-333233\tfrac13$ and $f(100)=333233\tfrac13$.
The smallest occurs at **x=−100**. Checking stationary points alone would miss the global endpoint minimum.