For the function $f(x) = x^{40} - x^{20}$,find the absolute minimum value in the interval $[0, 1]$.

The maximum value of $\frac{\log x}{x}$ is

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

If $f(x)=x^2+ax+b$ has a minima at $x=3$ whose value is $5$,then the values of $a$ and $b$ are respectively.

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,

The shortest distance of the point $(0,0)$ from the curve $y = e^x + e^{-x}$ is-