The function $f(x) = x^x$ is increasing when:

The function $f(x) = \frac{\log(\pi + x)}{\log(e + x)}$ is:

Find the intervals in which the function $f$ given by $f(x) = x^{2} - 4x + 6$ is:
$(a)$ increasing
$(b)$ decreasing

For the function $f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right)$,where $x \in\left[0, \frac{\pi}{2}\right]$,consider the following two statements:
$(I)$ $f$ is increasing in $\left(0, \frac{\pi}{2}\right)$.
$(II)$ $f^{\prime}$ is decreasing in $\left(0, \frac{\pi}{2}\right)$.
Which of the following is correct?

If $f(x) = x^3 - 6x^2 + 9x + 3$ is a decreasing function,then in which interval does $x$ lie?

If $f(x) = \int_x^{x+1} e^{-t^2} dt$, then the interval in which $f(x)$ is decreasing is