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

Explanation

(d) `f(x) = x^(100)+sin x-1`
` rArrf'(x) = 100x^(99) + cos x`
(a) In (0, 1),
` cos x gt 0`
`100x^(99) gt 0`
` 100x^(99) + cos x gt 0`
` rArr f'(x) gt 0`
`:. F(x)` is strictly increasing in (0, 1).
(b) In `(pi/2 , pi)`,
`cos x lt 0` ,
` 100x^(99) gt 0`
and ` 100x^(99) gt cos x`
`:. 100x^(99) - cos x gt 0`
` rArr f'(x) gt 0`
`:. f(x)" is strictly increasing in "(0,pi/2)`.