UPSC NDA I 2015 Mathematics Paper

Concept:Use the second derivative test to find the local maximum of a polynomial function.Explanation:Given $f(x) = 0.75x^{4} - x^{3} - 9x^{2} + 7$.Find the first derivative: $f'(x) = 3x^{3} - 3x^{2} - 18x$.Set $f'(x) = 0$ and solve: $3x(x^{2} - x - 6) = 0$ gives $3x(x-3)(x+2) = 0$.Thus critical points are $x = 0$, $x = 3$, and $x = -2$.Find the second derivative: $f''(x) = 9x^{2} - 6x - 18$.Evaluate $f''$ at each critical point:$f''(0) = -18 < 0$, so $x=0$ gives a local maximum.$f''(3) = 45 > 0$, so $x=3$ gives a local minimum.$f''(-2) = 30 > 0$, so $x=-2$ gives a local minimum.Therefore the maximum value occurs at $x = 0$.Compute $f(0) = 0.75(0)^{4} - (0)^{3} - 9(0)^{2} + 7 = 7$.Thus the maximum value of the function is $7$.Answer:$7$