GATE 2019 CE (CE1) – Question 36

Engineering Mathematics · Calculus: Limits, continuity, mean value theorems, maxima and minima, Taylor series · 2 marks · Multiple choice

Which statement is NOT correct?

$x^{1/x}$, x > 0, has its global maximum at x = e.
$x^{1/x}$, x > 0, has its global minimum at x = e.
$x^3$ has neither global minimum nor global maximum.
$|x|$ has its global minimum at x = 0.

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Correct answer: (B) $x^{1/x}$, x > 0, has its global minimum at x = e.

Explanation

Let $g(x)=x^{1/x}$. Then $\ln g=(\ln x)/x$ and $d(\ln g)/dx=(1-\ln x)/x^2$. It is positive below e and negative above e, giving a global maximum at e. Thus the global-minimum statement is false: **B**. The cubic is unbounded in both directions and absolute value has its global minimum at zero.