UPSC NDA I 2021 Mathematics Solved Paper

Concept:The slope of a curve is given by its first derivative. To find where the slope is maximum, we find the critical points of the slope function (where its derivative is zero) and then confirm that the second derivative is negative.Explanation:Let $y = -x^{3} + 3x^{2} + 2x - 27$.The slope $m = \frac{dy}{dx} = -3x^{2} + 6x + 2$.To maximize $m$, set $\frac{dm}{dx} = 0$.$\frac{dm}{dx} = -6x + 6 = 0 \Rightarrow x = 1$.Check the second derivative: $\frac{d^{2}m}{dx^{2}} = -6 < 0$, which confirms a maximum at $x = 1$.Answer:$x = 1$ (Option C)