JEE Mathematics 2024 Question with Solution

Given: $f(x) = 2x + 3x^{2/3}$ with $x \in \mathbb{R}$.

Find: The number of points of local maxima and local minima.

From the solution working, differentiate the function:

Set the derivative equal to zero to get stationary points:

Now check the second derivative shown in the solution:

At $x = -1$,

So $x = -1$ is a point of local maximum.

The solution also analyzes the sign of $f'(x)$ around $x = 0$ and shows the sign pattern changing from negative to positive there. Hence, $x = 0$ is a point of local minimum.

Therefore, the function has exactly one point of local maximum and exactly one point of local minimum. The correct option is C.

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

Find: Whether the function has local maxima or minima.

Using the derivative from the provided solution:

The derivative is not defined at $x = 0$, so $x = 0$ must also be checked as a critical point.

From the sign chart in the solution:

• $f'(x) > 0$ for $x < -1$
• $f'(x) < 0$ for $-1 < x < 0$
• $f'(x) > 0$ for $x > 0$

So the function changes from increasing to decreasing at $x = -1$, which gives a local maximum, and from decreasing to increasing at $x = 0$, which gives a local minimum.

Hence there is exactly one local maximum and exactly one local minimum. The correct option is C.