If `f:[0,1] to [0,prop)` is differentiable function with decreasing first derivative suc that f(0)=0 and `f'(x) gt 0`, then

Options

A. `underset(0)overset(1)int (1)/(f^(2)(x)+1)dx gt(f(1))/(f'(1))`
B. `underset(0)overset(1)int (1)/(f^(2)(x)+1)dx lt(f(1))/(f'(1))`
C. `underset(0)overset(1)int (1)/(f^(2)(x)+1)dx lt(tan^(-1)(f(1)))/(f'(1))`
D. `underset(0)overset(1)int (1)/(f^(2)(x)+1)dx =(f(1))/(f'(1))`

Answer

C

Explanation

It is given that f'(x) is decreasing on [0,1]. Therefore, f'(1) is the least value of f'(x) in [0,1].
`:. F'(x) ge f'(1)` for all `x in [0,1]`
`rArr (f'(x))/(f^(2)(x)+1) ge (f'(1))/(f^(2)(x)+1)`
`rArr underset(0)overset(1)int (f(x))/(f^(2)(x)+1)dx ge underset(0)overset(1)int (f'(1))/(f^(2)(x)+1)dx`
`rArr [tan^(-1)(f(x))]_(0)^(1) ge underset(0)overset(1)int (f'(1))/(f^(2)(x)+1)dx`
`rArr tan^(-1)(f(1))-tan^(-1)(f(0)) ge underset(0)overset(1)int (f'(1))/(f^(2)(x)+1)dx`
`rArr tan^(-1)(f(1)) ge underset(0)overset(1)int (f'(1))/(f^(2)(x)+1)dx`