JEE Main 2025 — Application Of Derivatives Question with Solution

We are given

$$f(x)={\int }_0^{x^2}\frac{t^2-8t+15}{e^t}dt,x\in \mathbb{R}.$$

To find the local extrema, we first compute the derivative using the Fundamental Theorem of Calculus and the chain rule.

Step 1. Rewrite the derivative:

Let

$$F(u)={\int }_0^u\frac{t^2-8t+15}{e^t}dt,$$

so that

$$f(x)=F(x^2).$$

Then by the chain rule,

$$f^{\prime }(x)=F^{\prime }(x^2)\cdot 2x.$$

Since

$$F^{\prime }(u)=\frac{u^2-8u+15}{e^u},$$

we substitute $u=x^2$ to get

$$f^{\prime }(x)=\frac{(x^2{)}^2-8x^2+15}{e^{x^2}}\cdot 2x=\frac{2x(x^4-8x^2+15)}{e^{x^2}}.$$

Step 2. Factor and Identify Critical Points:

Factor the polynomial $x^4-8x^2+15$ by writing it in terms of $x^2$. Let $y=x^2$, then

$$y^2-8y+15=(y-3)(y-5)$$

so that

$$x^4-8x^2+15=(x^2-3)(x^2-5).$$

Thus,

$$f^{\prime }(x)=\frac{2x(x^2-3)(x^2-5)}{e^{x^2}}.$$

Since $e^{x^2}>0$ for all $x$, the zeros of $f^{\prime }(x)$ are determined by

$$2x(x^2-3)(x^2-5)=0.$$

That gives the critical points:

$2x=0\Rightarrow x=0$,

$x^2-3=0\Rightarrow x=\pm \sqrt{3}$,

$x^2-5=0\Rightarrow x=\pm \sqrt{5}$.

Step 3. Analyzing the Sign of $f^{\prime }(x)$:

We need to determine the nature (maximum or minimum) by looking at the sign changes of $f^{\prime }(x)$ on the intervals determined by the critical points $x=-\sqrt{5}$, $x=-\sqrt{3}$, $x=0$, $x=\sqrt{3}$, and $x=\sqrt{5}$. Notice that the factor $2x(x^2-3)(x^2-5)$ will dictate the sign.

Let’s define:

$$h(x)=x(x^2-3)(x^2-5).$$

Examine the sign of $h(x)$ in each interval:

For $x<-\sqrt{5}$:

$x$ is negative.

$x^2>5$ so $x^2-3>0$ and $x^2-5>0$.

Product: negative $\times$ positive $\times$ positive = negative.

Thus, $f^{\prime }(x)<0$.

For $-\sqrt{5}<x<-\sqrt{3}$:

$x$ is negative.

$x^2$ is between 3 and 5 so $x^2-3>0$ while $x^2-5<0$.

Product: negative $\times$ positive $\times$ negative = positive.

Thus, $f^{\prime }(x)>0$.

For $-\sqrt{3}<x<0$:

$x$ is negative.

$x^2<3$ so both $x^2-3<0$ and $x^2-5<0$.

Product: negative $\times$ negative $\times$ negative = negative.

Thus, $f^{\prime }(x)<0$.

For $0<x<\sqrt{3}$:

$x$ is positive.

$x^2<3$ so both $x^2-3<0$ and $x^2-5<0$.

Product: positive $\times$ negative $\times$ negative = positive.

Thus, $f^{\prime }(x)>0$.

For $\sqrt{3}<x<\sqrt{5}$:

$x$ is positive.

$x^2$ is between 3 and 5 so $x^2-3>0$ while $x^2-5<0$.

Product: positive $\times$ positive $\times$ negative = negative.

Thus, $f^{\prime }(x)<0$.

For $x>\sqrt{5}$:

$x$ is positive.

$x^2>5$ so both $x^2-3>0$ and $x^2-5>0$.

Product: positive $\times$ positive $\times$ positive = positive.

Thus, $f^{\prime }(x)>0$.

Step 4. Classify the Critical Points:

At $x=-\sqrt{5}$: $f^{\prime }(x)$ changes from negative to positive → local minimum.

At $x=-\sqrt{3}$: $f^{\prime }(x)$ changes from positive to negative → local maximum.

At $x=0$: $f^{\prime }(x)$ changes from negative to positive → local minimum.

At $x=\sqrt{3}$: $f^{\prime }(x)$ changes from positive to negative → local maximum.

At $x=\sqrt{5}$: $f^{\prime }(x)$ changes from negative to positive → local minimum.

Step 5. Conclusion:

There are local maximum points at $x=-\sqrt{3}$ and $x=\sqrt{3}$ (2 points in total), and local minimum points at $x=-\sqrt{5}$, $x=0$, and $x=\sqrt{5}$ (3 points in total).

Thus, the numbers of local maximum and local minimum points of $f$ are 2 and 3, respectively.

The correct option is Option C.