On the interval $[0,1]$,the function $f(x) = x^{25}(1-x)^{75}$ takes its maximum value at the point

Maximum value of $x(1 - x)^2$ when $0 \le x \le 2$,is

For the function $f(x) = x \cos \frac{1}{x}, \quad x \geq 1$,consider the following statements:
$(A)$ For at least one $x$ in the interval $[1, \infty), f(x+2)-f(x) < 2$
$(B)$ $\lim _{x \rightarrow \infty} f^{\prime}(x) = 1$
$(C)$ For all $x$ in the interval $[1, \infty), f(x+2)-f(x) > 2$
$(D)$ $f^{\prime}(x)$ is strictly decreasing in the interval $[1, \infty)$
Which of the following combinations of statements is correct?

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:

For all $x \in (0, 1)$,which of the following inequalities is true?

If $f(x) = 1 + 2 \sin x + 3 \cos^2 x$ for $0 < x < 2\pi / 3$,then: