For which values of $x$ is the function $f(x) = \sin x + \cos 2x$ $(x > 0)$ minimum?

If $x + y = 8$,find the maximum value of $xy$.

If $x+y=60$,$x>0$,$y>0$,then the maximum value of $x y^3$ is

The function,$f(x)=x \sqrt{1-x}$,where $x \in(0,1)$,has a local maximum at $x=$

The minimum value of the function $f(x) = 3x^4 - 8x^3 + 12x^2 - 48x + 25$ on the interval $[0, 3]$ is equal to:

Let a function $f(x)$ be continuous in an interval $[a, b]$. Let $\delta > 0$ be a very small real number. Let $c \in (a, b)$ be such that $f(c - \delta) < f(c)$ and $f(c + \delta) < f(c)$ for every $\delta > 0$. Let $(f(\alpha - \delta) - f(\alpha))(f(\alpha + \delta) - f(\alpha)) < 0$ for all $\alpha \in (a, b)$ and $\alpha \neq c$. Then: