In the open interval $\left(0, \frac{\pi}{2}\right)$,which of the following is true for the expression $\cos x + x \sin x$?

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

If $f(x) = x^3 + bx^2 + cx + d$ and $0 < b^2 < c$,then on $R$,$f(x)$ is:

The maximum interval in which the slopes of the tangents drawn to the curve $y=x^4+5x^3+9x^2+6x+2$ increase is

The function $f(x) = ax + b$ is strictly increasing for all real $x$ if

Consider two statements $S_1$ and $S_2$.
$S_1$: If $f(x)$ is a differentiable function with $f'(x) > 0$ in $(a, b)$ and $f(x)$ is increasing in $(a, b)$,then $\frac{f(x)}{f'(x)}$ is also increasing in $(a, b)$.
$S_2$: Both $\sin x$ and $\tan x$ are increasing functions in $(0, \frac{\pi}{2})$.
Which of the following is true?