Question

The set of critical points of the function

f(x) = x – lnx + $\int_{2}^{x}{\left( \frac{1}{t} - 2 - 2\cos{}4t \right)dt}$, n Ī N

• A. $\left\{ \frac{n\pi}{2} \right\}$
• B. $\left\{ \frac{n\pi}{2} \pm \frac{\pi}{6} \right\}$
• C. $\left\{ \frac{\pi}{6},\frac{n\pi}{2} \pm \frac{\pi}{6} \right\}$
• D. None

Answer 1

Answer: (C)

$\left\{ \frac{\pi}{6},\frac{n\pi}{2} \pm \frac{\pi}{6} \right\}$

Answer 2

f ¢(x) = 1 – $\frac{1}{x}$ + $\left( \frac{1}{x} - 2 - 2\cos 4x \right)$

= –1 –2 cos 4x = 0 Ž cos 4x = –1/2

Ž x =$\frac{n\pi}{2}$+ $\frac{\pi}{6}$, n Ī I

but for log x, x > 0

for n = 0, x = ± p/6 (but x= –p/6 is not possible)

critical point = $\left\lbrack \frac{\pi}{6},\frac{n\pi}{2} \pm \frac{\pi}{6},n \in N \right\rbrack$