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

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:

The absolute minimum value of the function $f(x) = |x^2 - x + 1| + [x^2 - x + 1]$,where $[t]$ denotes the greatest integer function,in the interval $[-1, 2]$,is:

The minimum value of the slope of the tangent to the curve $y=x^3-3x^2+2x+93$ is

The sum of all the local minimum values of the twice differentiable function $f: R \rightarrow R$ defined by $f(x)=x^{3}-3 x^{2}-\frac{3 f^{\prime \prime}(2)}{2} x+f^{\prime \prime}(1)$ is:

Given that the solid obtained by rotating a rectangle about one of its sides is a cylinder. If the perimeter of a rectangle is $48 \text{ cm}$ and the volume of the cylinder formed by rotating it is maximum,then the dimensions of that rectangle are: