The GATE Grind

GATE 2025 DA – Question 49

Calculus and Optimization · Taylor series, maxima and minima, single-variable optimization · 2 marks · Multiple select

Consider the function $f(x) = \frac{x^3}{3} + \frac{7}{2}x^2 + 10x + \frac{133}{2}$, $x \in [-8, 0]$. Which of the following statements is/are correct?

  1. The maximum value of $f$ is attained at $x = -5$
  2. The minimum value of $f$ is attained at $x = -2$
  3. The maximum value of $f$ is $\frac{133}{2}$
  4. The minimum value of the derivative of $f$ is attained at $x = -\frac{7}{2}$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (C) The maximum value of $f$ is $\frac{133}{2}$; (D) The minimum value of the derivative of $f$ is attained at $x = -\frac{7}{2}$

Explanation

$f'(x) = x^2 + 7x + 10 = (x + 2)(x + 5)$, so $x = -5$ is a local maximum and $x = -2$ is a local minimum. The values are $f(-8) = 39.83$, $f(-5) = 62.33$, $f(-2) = 57.83$ and $f(0) = 66.5 = \frac{133}{2}$. The largest is $f(0) = \frac{133}{2}$, so the maximum is at the end point $x = 0$, not at $-5$ (A is false, C is true). The smallest is $f(-8) = 39.83$, not $f(-2)$ (B is false). The derivative $f'(x) = x^2 + 7x + 10$ is a parabola with its minimum where $f''(x) = 2x + 7 = 0$, which is $x = -\frac{7}{2}$ (D is true).