If $ f(x) = \int_{x}^{x^2} \frac{dt}{(logt)^2} , x \ne 0$ then f(x) is

Q: If $\displaystyle f(x) = \int_{x}^{x^2} \frac{dt}{(logt)^2} , x \ne 0$ then f(x) is

(A) monotonically increasing in (2 , ∞)

(B) monotonically decreasing in (1 , 2)

(C) monotonically decreasing in (2 , ∞)

(D) monotonically decreasing in (0 , 1)

Ans: (A) , (B)

Sol: $\displaystyle f'(x) = \frac{2x}{(logx^2)^2} – \frac{1}{(logx)^2}$

$\displaystyle f'(x) = \frac{x-2}{2(logx)^2} $

f'(x) > 0 for x > 2

and f'(x) < 0 for x < 2

Hence (A), (B) are correct answers.