GATE 2015 CS (CS2) – Question 56
Let $f(x)=x^{-1/3}$ and let A be the area between its graph and the x-axis from −1 to 1. Which are true? I: f is continuous on [−1,1]. II: f is unbounded on [−1,1]. III: A is nonzero and finite.
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Correct answer: (C) II and III only
Explanation
The function is undefined and unbounded at zero, so I is false and II true. Area uses absolute value: $A=2\int_0^1x^{-1/3}\,dx=2[\frac32x^{2/3}]_0^1=3$. The improper integral converges, proving III. Thus C.