NDA 2 2025 Maths, question 100

Topic: Differential calculus

Q100

Consider the following statements :

Statement-I : The function $f(x)=\frac{x^{3}+128}{x}$ has a minimum value $48$ at $x=4$ .

Statement-II : As $x$ increases through $4$ , $f'(x)$ changes sign from positive to negative.

Which one of the following is correct in respect of the above statements?

(a) Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
(b) Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-I
(c) Statement-I is correct but Statement-II is not correctCorrect answer
(d) Statement-I is not correct but Statement-II is correct

Solution.

Answer: (c) Statement-I is correct but Statement-II is not correct

$f(x) = x^{2} + \frac{128}{x}$ , so $f'(x) = 2x - \frac{128}{x^{2}} = \frac{2(x^{3} - 64)}{x^{2}}$ .
$f'$ goes from negative to positive at $x = 4$ , so $x = 4$ is a minimum with $f(4) = 16 + 32 = 48$ : I is correct.
II says the sign goes from positive to negative, which is wrong.

Trap: 48 is a local minimum (the least value for $x > 0$ ); over all $x \neq 0$ the function has no lowest value, since it falls without limit as $x \to 0^{-}$ .