Question

Let (2, 3) be the largest open interval in which the function $f(x) = 2\log_e(x - 2) - x^2 + ax + 1$ is strictly increasing and (b, c) be the largest open interval, in which the function $g(x) = (x - 1)^3(x + 2 - a)^2$ is strictly decreasing. Then 100(a + b - c) is equal to :

Answer 1

360

Answer 2

The derivative of $f(x)$ is $f'(x) = \frac{2}{x - 2} - 2x + a$. For $f(x)$ to be strictly increasing on $(2, 3)$, $f'(x) > 0$ for $x \in (2, 3)$ and $f'(3) = 0$. Setting $f'(3) = 0$ gives $\frac{2}{3-2} - 2(3) + a = 0 \implies 2 - 6 + a = 0 \implies a = 4$. With $a=4$, $g(x) = (x - 1)^3(x - 2)^2$. The derivative is $g'(x) = 3(x - 1)^2(x - 2)^2 + (x - 1)^3 \cdot 2(x - 2) = (x - 1)^2(x - 2)[3(x - 2) + 2(x - 1)] = (x - 1)^2(x - 2)(5x - 8)$. For $g(x)$ to be strictly decreasing, $g'(x) < 0$. Since $(x-1)^2 \ge 0$, this requires $(x - 2)(5x - 8) < 0$. The roots are $x=2$ and $x=8/5$. The quadratic $(x - 2)(5x - 8)$ is negative between its roots, so $x \in (8/5, 2)$. Thus, $b = 8/5$ and $c = 2$. The required value is $100(a + b - c) = 100(4 + 8/5 - 2) = 100(2 + 8/5) = 100(10/5 + 8/5) = 100(18/5) = 20 \times 18 = 360$.