GATE CS 2023 Q.28: Calculus

Let

f(x) = x^3 + 15x^2 - 33x - 36

be a real-valued function.

Which of the following statements is/are TRUE?

$f(x)$ does not have a local maximum.

$f(x)$ has a local maximum.

$f(x)$ does not have a local minimum.

$f(x)$ has a local minimum.

Official answer and solution

Solution

Find critical points using the first derivative and test their nature using the second derivative.

Given $f(x) = x^3 + 15x^2 - 33x - 36$ :

First derivative:
$f'(x) = 3x^2 + 30x - 33 = 3(x^2 + 10x - 11) = 3(x + 11)(x - 1)$
Setting $f'(x) = 0$ yields critical points at $x = -11$ and $x = 1$ .
Second derivative:

$f''(x) = 6x + 30$
- At $x = -11$ : $f''(-11) = 6(-11) + 30 = -36 < 0$ , which signifies a local maximum at $x = -11$ .
- At $x = 1$ : $f''(1) = 6(1) + 30 = 36 > 0$ , which signifies a local minimum at $x = 1$ .

Therefore, $f(x)$ has a local maximum (statement B is TRUE) and $f(x)$ has a local minimum (statement D is TRUE).

Final answer

Options (B, D)

Common trap

Selecting only one extremum or confusing the signs of $f''(x)$ ( $f'' < 0$ implies local maximum).

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