Let $f(x) = x^3 + bx^2 + cx + d$ where $0 < b^2 < c$. Then $f(x)$:

The function $f(x) = (1/2)^x$ on $R$ is

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,

The function $f(x)=\log (1+x)-\frac{2 x}{2+x}$ is increasing on

On which of the following intervals is the function $f$ given by $f(x)=x^{100}+\sin x-1$ decreasing?

If $f(x) = xe^{x(1-x)}$,then $f(x)$ is...