If the Lagrange' mean value theorem is applied to the function $f(x)=e^x$ defined on the interval $[1,2]$ and the value of $c \in(1,2)$ is $k$, then $e^{k-1}=$

Consider the quadratic equation $a x^2+b x+c=0$, where $2 a+3 b+6 c=0$ and let $g(x)=\frac{a x^3}{3}+\frac{b x^2}{2}+c x$

Statement I The given quadratic equation $a x^2+b x+c=0$ has atleast one root in $(0,1)$.

Statement II Rolle's theorem is applicable to $g(x){\text {on }}$ [0, 1].

Then

The difference between the absolute maximum and absolute minimum values of the function $f(x)=2 x^3-15 x^2+36 x-30$ on $[-1,4]$ is

If $f(x)=x e^{x(1-x)}, x \in R$, then $f(x)$ is