Let $\phi(x) = f(x) + f(2a - x)$, $x \in [0, 2a]$ and $f^{\prime \prime}(x) > 0$ for all $x \in [0, a]$. Then $\phi(x)$ is

The function $f(x) = [x(x-2)]^2$ is increasing in the set

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 $[\frac{\pi}{2}, \pi]$.

What is the length of the longest interval in which the function $f(x) = 3\sin x - 4\sin^3 x$ is an increasing function?

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

For which value of $x$ is the function $f(x) = x^2 - 2x$ decreasing?