Mean Value Theorem

Verify the Cauchy's mean value theorem for the functions

$f\left(x\right)=\frac{1}{x}$, and $g\left(x\right)=x^2-4$ on the interval $\left[1,2\right]$.

Check the validity of Cauchy's mean value theorem for the functions

$f\left(x\right)=x^3$, and $g\left(x\right)=x^2$ on the interval $\left(0,2\right)$.

Verify Cauchy's mean value theorem for the functions $f\left(x\right)=\sin x$, and $g\left(x\right)=\cos x$ in $\left[0,\frac{\mathrm{\pi }}{2}\right]$.

Check the validity of Cauchy's mean value theorem for the functions

$f\left(x\right)=x^4$, and $g\left(x\right)=x^2$ on the interval $\left[1,2\right]$.

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

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 

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

If $f\left(x\right)=\log \left(\sin x\right), x\in \left[\frac{\pi }{6},\frac{5\pi }{6}\right],$ then value of $c$ by applying $\mathrm{L}.\mathrm{M}.\mathrm{V}.\mathrm{T}.$ is

The constant $c$ of Lagrange's mean value theorem for the function $f\left(x\right)=\frac{2x+3}{4x-1}$ defined on $\left[1,2\right]$ is

Let $f\left(x\right)$ be differentiable on $[1,6]$ and $f\left(1\right)=-2 .$ If $f\left(x\right)$ has only one root in $(1,6)$, then there exists $c\in (1,6)$ such that

Find the value of ${}^{\text{'}}p^{\text{'}}$ and ${}^{\text{'}}q^{\text{'}}$ if the function $f\left(t\right)=t^3-6t^2+pt+q$ defined on $\left[1,3\right]$ satisfies the Rolle's theorem for $c=\frac{2\sqrt{3}+1}{\sqrt{3}}$

Let $g\left(t\right)={\int }_{x_1}^{x_2}f\left(t, x\right)dx$. Then $g\text{'}\left(t\right)={\int }_{x_1}^{x_2}\frac{\partial }{\partial t}\left(f\left(t, x\right)\right)dx$. Consider $f\left(x\right)={\int }_0^{\mathrm{\pi }}\frac{\ln \left(1+x\cos \theta \right)}{\cos \theta }d\theta$.

$f\left(x\right)$ is

The value of $\text{'}c\text{'}$ when Cauchy's mean value theorem is applied for the functions $f\left(x\right)=\cos x \& g\left(x\right)=\sin x$ in the interval $\left[a,b\right]$ is $c=\frac{a+b}{m}$. Then the value of $m$ is

Cauchy's mean value theorem is valid for the functions $f\left(x\right)=x^4 \& g\left(x\right)=x^2$ in the interval $\left[1,2\right]$

$\forall x\ne 0, 1-\frac{x^2}{2}<\cos x$

Verify L.M.V.T. for the function

$f\left(x\right)=x{\left(x+4\right)}^2, x\in \left[0,4\right]$

Verify Rolle's theorem for the following function

$f\left(x\right)=e^{-x}\left(\sin x-\cos x\right) x\in \left[\frac{\mathrm{\pi }}{4},\frac{5\mathrm{\pi }}{4}\right]$

If Rolle's theorem holds for the function $f\left(x\right)=(x-2)\log x,x\in [1,2],$ show that the equation $x\log x=2-x$ is satisfied by at least one value of $x$ in (1,2) .

Find $c$ if LMVT is applicable for
$f\left(x\right)=x(x-1)(x-2), x\in [0,1/2]$