JEE Mathematics 2024 Question with Solution

Given: $f(x)=\begin{cases}x^2+3x+a, & x\leq 1\\ bx+2, & x>1\end{cases}$ is differentiable on $\mathbb{R}$.

Find: The value of $\int_{-2}^{2} f(x) \, dx$.

For differentiability at $x=1$, the function must be continuous and have equal left and right derivatives.

From continuity at $x=1$:

$$4+a=b+2$$

So,

$$b-a=2$$

From differentiability at $x=1$:

$$\frac{d}{dx}(x^2+3x+a)=2x+3$$

Hence the left derivative at $x=1$ is

$$f'_-(1)=2(1)+3=5$$

For $x>1$,

$$\frac{d}{dx}(bx+2)=b$$

So the right derivative at $x=1$ is

$$f'_+(1)=b$$

Therefore,

$$b=5$$

Using $b-a=2$,

$$5-a=2$$ $$a=3$$

Thus,

$$f(x)=\begin{cases}x^2+3x+3, & x\leq 1\\ 5x+2, & x>1\end{cases}$$

Now split the integral at $x=1$:

$$\int_{-2}^{2} f(x) \, dx = \int_{-2}^{1} (x^2+3x+3) \, dx + \int_{1}^{2} (5x+2) \, dx$$

For the first part,

$$\int_{-2}^{1} (x^2+3x+3) \, dx = \left[\frac{x^3}{3}+\frac{3x^2}{2}+3x\right]_{-2}^{1} = \frac{15}{2}$$

For the second part,

$$\int_{1}^{2} (5x+2) \, dx = \left[\frac{5x^2}{2}+2x\right]_{1}^{2} = \frac{19}{2}$$

Adding,

$$\int_{-2}^{2} f(x) \, dx = \frac{15}{2}+\frac{19}{2}=17$$

Therefore, the value of the integral is $17$, so the correct option is D.

Given: The function changes definition at $x=1$.

Find: Use continuity and differentiability to determine $a$ and $b$, then evaluate the integral.

1. Continuity at $x=1$ gives

$$\lim_{x\to 1^-} f(x)=\lim_{x\to 1^+} f(x)$$

That is,

$$1^2+3(1)+a=b(1)+2$$ $$4+a=b+2$$

2. Differentiability at $x=1$ gives equality of one-sided derivatives:

$$f'_-(1)=f'_+(1)$$

Since

$$f'(x)=2x+3 \quad \text{for} \quad x<1$$

we get

$$f'_-(1)=5$$

And for the second part,

$$f'(x)=b \quad \text{for} \quad x>1$$

so

$$f'_+(1)=b$$

Hence,

$$b=5$$

Substituting into the continuity equation,

$$4+a=7$$ $$a=3$$

3. Evaluate the integral:

$$\int_{-2}^{2} f(x) \, dx = \int_{-2}^{1} (x^2+3x+3) \, dx + \int_{1}^{2} (5x+2) \, dx$$

The first integral equals $\frac{15}{2}$ and the second equals $\frac{19}{2}$. Thus,

$$\frac{15}{2}+\frac{19}{2}=17$$

Therefore, the correct option is D.