Find Difference Between Maximum and Minimum Values of Integral Function

To find the value of $3(M - m)$, we proceed step-by-step:

Step 1: Find the value of $\alpha$ Given that matrix $A$ is singular, its determinant must be equal to zero, i.e., $\det(A) = 0$.

$$A = \begin{bmatrix} 1 & 3 & -1 \\ 2 & 1 & \alpha \\ 0 & 1 & -1 \end{bmatrix}$$

Expanding the determinant along the first column:

$$\det(A) = 1 \cdot \begin{vmatrix} 1 & \alpha \\ 1 & -1 \end{vmatrix} - 2 \cdot \begin{vmatrix} 3 & -1 \\ 1 & -1 \end{vmatrix} + 0 \cdot \begin{vmatrix} 3 & -1 \\ 1 & \alpha \end{vmatrix}$$ $$\det(A) = 1((-1) - \alpha) - 2(-3 - (-1)) + 0$$ $$\det(A) = -1 - \alpha - 2(-2) = 3 - \alpha$$

Since $\det(A) = 0$:

$$3 - \alpha = 0 \implies \alpha = 3$$

Thus, the interval for $x$ is $[1, \alpha] = [1, 3]$.

---

Step 2: Determine the nature of $f(x)$ on $[1, 3]$ The function $f(x)$ is given by:

$$f(x) = \int_{0}^{x} (t^2 + 2t + 3) \, dt$$

Using the Leibniz Rule for differentiation under the integral sign:

$$f'(x) = x^2 + 2x + 3 = (x + 1)^2 + 2$$

Since $(x + 1)^2 + 2 > 0$ for all real $x$, $f'(x) > 0$ on the interval $[1, 3]$. This implies that $f(x)$ is a strictly increasing function on $[1, 3]$.

---

Step 3: Find the maximum ($M$) and minimum ($m$) values Since $f(x)$ is strictly increasing on $[1, 3]$:

• The minimum value $m$ occurs at the lower boundary, $x = 1$, so $m = f(1)$.
• The maximum value $M$ occurs at the upper boundary, $x = 3$, so $M = f(3)$.

Therefore, the difference $M - m$ is:

$$M - m = f(3) - f(1) = \int_{0}^{3} (t^2 + 2t + 3) \, dt - \int_{0}^{1} (t^2 + 2t + 3) \, dt = \int_{1}^{3} (t^2 + 2t + 3) \, dt$$

---

Step 4: Evaluate the integral

$$M - m = \left[ \frac{t^3}{3} + t^2 + 3t \right]_{1}^{3}$$ $$M - m = \left( \frac{3^3}{3} + 3^2 + 3(3) \right) - \left( \frac{1^3}{3} + 1^2 + 3(1) \right)$$ $$M - m = (9 + 9 + 9) - \left( \frac{1}{3} + 1 + 3 \right)$$ $$M - m = 27 - \frac{13}{3} = \frac{81 - 13}{3} = \frac{68}{3}$$

---

Step 5: Compute $3(M - m)$

$$3(M - m) = 3 \times \frac{68}{3} = 68$$

Correct Option: B