If $f(x) = \frac{1}{x + 1} - \log(1 + x)$,where $x > 0$,then $f$ is:

Let $f(x) = xe^{x(1 - x)}$. Then $f(x)$ is:

$A$ function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?
Interval | Function

For the function $f(x) = \cos x - x + 1, x \in R$,consider the following two statements:
$(S1)$ $f(x) = 0$ for only one value of $x$ in $[0, \pi]$.
$(S2)$ $f(x)$ is decreasing in $[0, \frac{\pi}{2}]$ and increasing in $[\frac{\pi}{2}, \pi]$.

Prove that the function given by $f(x) = \cos x$ is decreasing in $(0, \pi)$.

Let $f : R \rightarrow R$ be a twice differentiable function such that $f^{\prime\prime}(x) > 0$ for all $x \in R$ and $f^{\prime}(a-1) = 0$,where $a$ is a real number. Let $g(x) = f(\tan^{2}x - 2\tan x + a)$,$0 < x < \frac{\pi}{2}$. Consider the following two statements:
$(I)$ $g$ is increasing in $(0, \frac{\pi}{4})$
$(II)$ $g$ is decreasing in $(\frac{\pi}{4}, \frac{\pi}{2})$
Then,