The function `x^x` decreases in the interval `(0,e)` (b) `(0,1)` `(0,1/e)` (d) none of these

Options

A. `(0,1)`
B. `(-1,0)`
C. `(1,e)`
D. none of these

Answer

C

Explanation

Since `e^(x^(2))` is an increasing function on `(0,1)`.
`m=e^(0)=1,M=e^(1)=e` [`m` and `M` are minimum and maximum values of `f(x)=e^(x^(2))` in the interval `(0,1)` respectively]. Then
`1lt e^(x^(2)) lt e`, for all `x epsilon(0,1)`
or `1(1-0)lt int_(0)^(1) e^(x^(2))dx lt e(1-0)`
or `1 lt int_(0)^(1)e^(x^(2)) dx lt e`