On which of the following intervals is the function `x^(100)+sinx-1` decreasing? `(,pi/2)` (b) `(0,1)` `(pi/2,pi)` (d) none of these

Options

A. `(0,pi//2)`
B. (0,1)
C. `(pi//2,pi)`
D. None of these

Answer

4

Explanation

`f(X)=x^(100)+sinx-1`
`therefore f(X)=100x^(99)+cosx`
if `0ltxlt(pi)/(2) , "then" f(X)gt0`
Therefore f(X) is increasing on `(0,pi//2)`
If `0ltxlt(pi)/(2) "then" f(X)gt0`
Therefore f(X) is increasing on `(0,pi//2)`
If `0ltxlt(pi)/(2)`
If `0ltxlt1` then
`100x^(99)gt0 and cos x gt0`
Thus f(X) is increasing on (0,1)
if `(pi)/(2)ltxltpi` then
`100x^(99)gt100`
or `100x^(99)+cosxgt0`
`f(x) gt0` implies f(X) is increasing in `(pi//2,pi)`