The maximum value of ${x^{1/x}}$ is

Let $f(x) = x^2 e^{-2x}, x > 0$. The maximum value of $f(x)$ is

Find the maximum and minimum values of the function $f(x)=(2x-1)^{2}+3$.

The maximum area of a right-angled triangle with hypotenuse $h$ is

If the function $f(x) = \frac{t + 3x - x^2}{x - 4}$,where $t$ is a parameter,has a local maximum and a local minimum,then the range of values of $t$ is:

Let $f :[2,4] \rightarrow R$ be a differentiable function such that $(x \ln x) f'(x) + (\ln x + 1) f(x) \geq 1$ for all $x \in [2,4]$,with $f(2) = \frac{1}{2}$ and $f(4) = \frac{1}{4}$. Consider the following two statements:
$(A): f(x) \leq 1$ for all $x \in [2,4]$
$(B): f(x) \geq \frac{1}{8}$ for all $x \in [2,4]$
Then,