Mean Value Theorems

Function $f\left(x\right)=\sqrt{1-x^2}$ does not satisfies Rolle’s theorem in $\left[-1,1\right]$.

The point on the curve $y=x^2$, where the tangent is parallel to the line joining the points $(1, 1)$ and $(2, 4)$ is

Function $f\left(x\right)=\left(x-1\right){\left(x-2\right)}^2$ satisfies Rolle's theorem in $\left[1,2\right]$.

Function $f\left(x\right)=x^3+3x^2-24x-80$ does not satisfies Rolle’s theorem in $\left[-4,5\right]$.

The $c$ value  of Lagrange’s mean-value theorem for $f\left(x\right)=\sqrt{25-x^2}$ on $\left[1,5\right]$ is $\sqrt{k}$. Find $k$.

Function $f\left(x\right)=(x-1)(x-2)(x-3)$ satisfies Rolle’s theorem in $\left[1,3\right]$.

Function $f\left(x\right)=2x^2-3x+1$ does not satisfies Lagrange’s mean-value theorem on $\left[1,3\right]$.

Lagrange’s mean-value theorem is not applicable on function $f\left(x\right)=\frac{1}{x}$ on $\left[-1,1\right]$.

Function $f\left(x\right)=x{\left(x-4\right)}^2$ does not satisfies Rolle’s theorem on $\left[0,4\right]$.

Function $f\left(x\right)=x^2+2x+3$ does not satisfies Lagrange’s mean-value theorem on $\left[4,6\right]$.

Function $f\left(x\right)=x^3+x^2-6x$ satisfies Lagrange’s mean-value theorem on $\left[-1,4\right]$.

The point on the curve $y=x^3-3x$, where the tangent to the curve is parallel to the chord joining $(1,-2)$ and $(2, 2)$ is 

Function $f\left(x\right) = x^3-7x^2+16x-12$ satisfies Rolle’s theorem in $\left[2,3\right]$.

Function $f\left(x\right)=e^{-x}\sin x$ does not satisfies Rolle’s theorem in $\left[0,\pi \right]$.

The $‘c’$ value  of Lagrange’s mean-value theorem for $\mathrm{f}\left(\mathrm{x}\right)=\sqrt{\mathrm{x}+2} \mathrm{on} \left[4,6\right]$ is $\frac{k}{2}+2\sqrt{k}$. Find $k$.

Function $f\left(x\right)=x^2-4x+3$ in satisfies Rolle's theorem $\left[1,3\right]$.

Function $f\left(x\right)=\sin x+\cos x$ in $\left[0,\frac{\pi }{2}\right]$ satisfies Rolle’s theorem.

Function $f\left(x\right)=x(x+2)e^x$ in $\left[-2,0\right]$ does not satisfies Rolle’s theorem.

The $c$ value  of Lagrange’s mean-value theorem for $f\left(x\right)=x^3-3x^2+2x$ on $\left[0,\frac{1}{2}\right]$ is equal to $1-\sqrt{\frac{k}{12}}$. Find $k$.