UPSC NDA 2 2016 Mathematics Solved Paper

Concept:A function is increasing if its derivative $f'(x) > 0$, and decreasing if $f'(x) < 0$.Explanation:For $-1 \le x \le 2$, $f(x) = 3x^2 + 12x - 1$.Differentiating: $f'(x) = 6x + 12$.At $x=1$, $f'(1) = 6(1)+12 = 18 > 0$, and since $f'(x)$ is linear, it is positive throughout $[-1,2]$.Hence $f(x)$ is increasing in $[-1,2]$ — statement 1 is correct.For $2 < x \le 3$, $f(x) = 37 - x$.Differentiating: $f'(x) = -1$, which is always negative.Hence $f(x)$ is decreasing in $(2,3]$ — statement 2 is correct.Answer:Both statements are correct. (Option C)