The value of $c$ in Rolle's theorem when $f\left(x\right)=2x^3-5x^2-4x+3$, is $x\in \left[\frac{1}{3},3\right]$

When the tangent to the curve $y=x\log x$ is parallel to the chord joining the points $(1,0)$ and $(e,e)$, the value of $x$ is

The value of $c$ in Lagrange's mean value theorem for the function $f\left(x\right)=x(x-2)$ when $x\in \left[1, 2\right]$ is

The value of $c$ in Rolle's theorem for the function $f\left(x\right)=x^3-3x$ in the interval $\left[0, \sqrt{3}\right]$ is 

If $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{e}}^{\mathrm{x}}\sin \mathrm{x} \mathrm{in} \left[\mathrm{o},\mathrm{\pi }\right],$ then $c$ in Rolle's theorem is