Question

The function $f(x) = 2\log(x - 2) - x^{2} + 4x + 1$ increases

in the interval

• A. (1,2)
• B. (2,3)
• C. $\left( \frac{5}{2},3 \right)$
• D. $(2,4)$

Answer 1

Answer: (B)

(2,3)

Answer 2

Note that $f(x)$ is defined for

$x > 2,f'(x) = 2/(x - 2) - 2(x - 2)$

= $2\left\lbrack 1 - (x - 2)^{2} \right\rbrack/(x - 2)$.

As $x > 2$, $f'(x) > 0,$ if $1 - (x - 2)^{2} > 0$ or $(x - 2)^{2}$< 1 or 1 < x < 3. As x > 2,

we get $f(x)$ increases on (2,3).