The difference between the maximum and minimum values of $f(x) = x^4e^{-x^2}$ for all $x \in R$ is:

If $xy = c^2$,what is the minimum value of $ax + by$ (where $a > 0, b > 0$)?

Let a function $f(x)$ be continuous in an interval $[a, b]$. Let $\delta > 0$ be a very small real number. Let $c \in (a, b)$ be such that $f(c - \delta) < f(c)$ and $f(c + \delta) < f(c)$ for every $\delta > 0$. Let $(f(\alpha - \delta) - f(\alpha))(f(\alpha + \delta) - f(\alpha)) < 0$ for all $\alpha \in (a, b)$ and $\alpha \neq c$. Then:

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

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?

Find both the maximum value and the minimum value of $3x^{4}-8x^{3}+12x^{2}-48x+25$ on the interval $[0,3]$.