If the function $f(x)=x^3+2 p x^2+27 x+16$ is strictly increasing for all $x \in R$,then the range of $p$ is

The function $f(x)=4 \sin ^3 x-6 \sin ^2 x+12 \sin x+100$ is strictly

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 $f(x)=2x^3-15x^2-144x-7$,then $f(x)$ is strictly decreasing in

Statement $-1:$ The function $f(x) = x^2(e^x + e^{-x})$ is increasing for all $x > 0.$
Statement $-2:$ The functions $g(x) = x^2e^x$ and $h(x) = x^2e^{-x}$ are increasing for all $x > 0$ and the sum of two increasing functions in any interval $(a, b)$ is an increasing function in $(a, b).$

The function $f(x) = x + \cos x$ is