At which point in the interval $[0, 1]$ is the function $f(x) = x^{25}(1 - x)^{75}$ maximum?

Find the local maximum and local minimum values for the function given by $g(x) = \frac{x}{2} + \frac{2}{x}, \quad x > 0$.

Of all the closed cylindrical cans (right circular) with a given volume of $100 \text{ cm}^3$,find the dimensions of the can that has the minimum surface area.

If the function $f$ is given by $f(x)=x^3-3(a-2)x^2+3ax+7$,for some $a \in R$,is increasing in $(0,1]$ and decreasing in $[1,5)$,then a root of the equation $\frac{f(x)-14}{(x-1)^2}=0$ $(x \neq 1)$ is

Let $f: R \rightarrow R$ be given by $f(x) = |x^{2} - 1|$, $x \in R$. Then:

The sum of absolute maximum and absolute minimum values of the function $f(x)=|2 x^{2}+3 x-2|+\sin x \cos x$ in the interval $[0,1]$ is