JEE Mathematics 2024 Question with Solution

Given: $f(x) = (x - 2)^{2/3}(2x + 1)$

Find: The number of critical points of $f(x)$.

A critical point occurs where $f'(x) = 0$ or where $f'(x)$ is undefined.

Using the product rule for $g(x) = (x - 2)^{2/3}$ and $h(x) = 2x + 1$,

where

So,

Taking the common factor $\frac{2}{3}(x - 2)^{-1/3}$,

Now simplify the bracket:

Hence,

Now find critical points:

1. For $f'(x) = 0$, the numerator must be zero.

2. The derivative is undefined at

because $(x - 2)^{1/3} = 0$ appears in the denominator.

Therefore, the function has two critical points, at $x = 1$ and $x = 2$. Hence, the correct option is A.

Given: $f(x) = (x - 2)^{2/3}(2x + 1)$

Find: Number of critical points.

After differentiating once and combining terms quickly,

So the derivative is zero at

and undefined at

Both belong to the domain of $f(x)$, so both are critical points.

Therefore, the number of critical points is $2$, so the correct option is A.