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

The sum of the maximum and minimum values of the function $f(x) = |5x - 7| + [x^2 + 2x]$ in the interval $[\frac{5}{4}, 2]$,where $[t]$ denotes the greatest integer function $\leq t$,is:

Let $f: R \rightarrow R$ be a function defined by $f(x) = (x - 3)^{n_{1}}(x - 5)^{n_{2}}$,where $n_{1}, n_{2} \in N$. Which of the following is $\text{NOT}$ true?

Let $f: D \rightarrow R$, $D \subseteq R$, $c \in D$ and $r$ be a non-zero real number. Consider the following statements:
$Y$. $c$ is an extreme point of $f \Rightarrow c$ is an extreme point of $rf$
$M$. $c$ is an extreme point of $f \Rightarrow c$ is an extreme point of $r+f$
Which of the following is correct?

The number of distinct real roots of $x^{4}-4x+1=0$ is

$A$ curve with equation of the form $y = ax^4 + bx^3 + cx + d$ has zero gradient at the point $(0, 1)$ and also touches the $x$-axis at the point $(-1, 0)$. Then the values of $x$ for which the curve has a negative gradient are: