Rolle's Theorem and Lagrange's Mean Value Theorem

Verify the conditions of the Lagrange's mean value theorem for the following function. In each case find a point $c$ in the interval as stated by the theorem $\log x$ on $[1,2]$

Verify the conditions of the Lagrange's mean value theorem for the following function. In each -case find a point $c$ in the interval as stated by the theorem $\sin x-\sin 2x$ on $[0,\pi ]$

Verify the conditions of the Lagrange's mean value theorem for the following function. In each case find a point $c$ in the interval as stated by the theorem $x^2-1$ on $\left[2,3\right]$

Verily the Rolle's theorem for the function $\left(x^2-1\right)(x-2)$ on $[-1,2]$.Find the point in the interval where the derivate vanishes.

Find $c$, so that $f^{\text{'}}\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$ in the following case $f\left(x\right)=e^x ; a=0, b=1$

Find $c$, so that $f^{\text{'}}\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$ in the following case $f\left(x\right)=x^2-3x-1;a=\frac{-11}{7}, b=\frac{13}{7}$

Find a point for the curve $y=x^3$, where the tangent is parallel to the chord joining $(1,1)$ and $(3,27)$.

Find a point on the curve $y=(x-3{)}^2$, where the tangent is parallel to the chord joining $(3,0)$and $(4,1)$.

Show that there is no real number $k$, for which the equation $x^2-3x+k=0$ has two distinct roots in $[0,1]$.

It is given that Rolle's theorem holds for the function $f\left(x\right)=x^3+b{x}^2+ax$ on $[1,3]$ with $c=2+\frac{1}{\sqrt{3}}$ Find the values of $a$ and $b$.

Verify Rolle's theorem for the following function $\log \left(x^2+2\right)-\log 3$ on $[-1,1]$

Verify Rolle's theorem for the following function $\sin x-\sin 2x$ on $[0,\pi ]$

Verify Rolle's theorem for the following function $x^2-1$ on $[-1,1]$.