Q.3. Given the following statements about a function ( f ). Find the value of ( a ).

• $f''(x) = a$ for all $x$
• $f(0) = 1$, $f(1) = 2$
• $\int_0^1 f(x) \, dx = 1$

Answer Choices

• $-6$
• $-3$
• $-2$
• $2$
• $3$
• $6$

Solution

We are given that $f''(x) = a$. Integrating this twice, we find the general form for $f(x)$:

$$f(x) = \frac{a}{2}x^2 + C_1x + C_2$$

Using the conditions $f(0) = 1$ and $f(1) = 2$, we solve for $C_1$ and $C_2$, which leads to:

$$C_2 = 1 \quad \text{and} \quad C_1 = 1 - \frac{a}{2}$$

Next, using the integral condition $\int_0^1 f(x) \, dx = 1$, we integrate and solve the equation:

$$\frac{a}{6} + \frac{1}{2} - \frac{a}{4} + 1 = 1$$

Simplifying this gives $a = 6$.

Thus, the value of $a$ is 6.