Applications Of Derivatives Question 3
Question: The function $ f(x)=2\log (x-2)-x^{2}+4x+1 $ increases on the interval
Options:
A) (1, 2)
B) (2, 3)
C) (1/2, 3)
D) (2, 4)
Show Answer
Answer:
Correct Answer: B
Solution:
[b] $ f(x)=2\log (x-2)-x^{2}+4x+1\Rightarrow f’(x) $
$ =\frac{2}{x-2}-2x+4 $
$ \Rightarrow f’(x)=2[ \frac{1-{{(x-2)}^{2}}}{x-2} ]=-2\frac{(x-1)(x-3)}{x-2} $
$ \Rightarrow f’(x)=\frac{2(x-1)(x-3)(x-2)}{{{(x-2)}^{2}}} $
$ \therefore f’(x)>0\Rightarrow -2(x-1)(x-3)(x-2)>0 $
$ \Rightarrow (x-1)(x-2)(x-3)<0\Rightarrow x\in (-\infty ,1)\cup (2,3) $ Thus, f(x) is increasing on $ (-\infty ,1)\cup (2,3) $ . Clearly, it includes answer [b] and (c).
BETA
