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

$y = 2x^3 - 8x^2 + 10x - 4$ is a function defined on $[1, 2]$. If the tangent drawn at a point $(a, b)$ on the graph of this function is parallel to the $X$-axis and $a \in (1, 2)$, then $a =$

Consider the function $f(x) = x(x - 1)(x - 2) \dots (x - 100)$. Which one of the following is correct?

Let $f(x)$ be a polynomial of degree four having extreme values at $x=1$ and $x=2$. If $\mathop {\lim }\limits_{x \to 0} \left[ {1 + \frac{{f(x)}}{{{x^2}}}} \right] = 3$,then $f(2)$ is equal to:

The sum of the maximum and the minimum values of $f(x) = 3x^4 - 2x^3 - 6x^2 + 6x + 4$ in the interval $(0, 2)$ is:

Find the local maximum and local minimum values of the function given by $f(x) = x^{3} - 6x^{2} + 9x + 15$.