GATE 2025 EC – Question 25
Consider the function $f:\mathbb{R}\to\mathbb{R}$, defined as
$$f(x)=2x^3-3x^2-12x+1.$$
Which of the following statements is/are correct?
(Here, $\mathbb{R}$ is the set of real numbers.)
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Show answer and explanation
Correct answer: (A) $f$ has no global maximizer; (B) $f$ has no global minimizer
Explanation
$f'(x)=6x^2-6x-12=6(x-2)(x+1)$ and $f''(x)=12x-6$. $f''(-1)=-18<0$, so $x=-1$ is a local maximizer; $f''(2)=18>0$, so $x=2$ is a local minimizer. So C and D are false. A cubic with a positive leading coefficient tends to $\pm\infty$ at $\pm\infty$, so it has neither a global maximum nor a global minimum: A and B are true.