Let $g(x) = 2f(x/2) + f(1 - x)$ and $f''(x) < 0$ for $0 \le x \le 1$. Then $g(x)$:

The interval in which the function $f(x) = x^{3} - 6x^{2} + 9x + 10$ is increasing is:

If the maximum value of $a$,for which the function $f_{a}(x)=\tan ^{-1} 2 x-3 a x+7$ is non-decreasing in $\left(-\frac{\pi}{6}, \frac{\pi}{6}\right)$,is $\bar{a}$,then $f_{\bar{a}}\left(\frac{\pi}{8}\right)$ is equal to

If the function $y = \sin x(1 + \cos x)$ is defined in the interval $[-\pi, \pi]$,then $y$ is strictly increasing in the interval

The value of $a$ such that $f(x) = \sqrt{3} \sin x - \cos x - 2ax + b$ decreases for all real values of $x$ is given by:

If the function $f(x) = \frac{K\sin x + 2\cos x}{\sin x + \cos x}$ is increasing for all values of $x,$ then