Mean Value Theorems

Show that between any two roots of the equation $e^x\cos x=1$ there exist atleast one root of $e^x\sin x-1=0$.

The second derivative of function exists for all $x\in \left[0,1\right]$ such that $\left|f^{"}\left(x\right)\right|\le 1$. If $f\left(0\right)=f\left(1\right)$, then show that $\left|f^{\text{'}}\left(x\right)\right|<1$ for all $x\in \left[0,1\right]$.

Let $f:\left[a,b\right]\to \left[a,b\right]$ (where $a<b$ and real numbers) be a differentiable non-linear function for which $f\left(a\right)=b$ and $f\left(b\right)=a$. Then

Let $f\left(x\right)$ be a function, continuous on $\left[a,b\right]$ and twice differentiable on $\left(a,b\right)$ consider the function $\varphi \left(x\right)=f\left(b\right)-f\left(x\right)-\left(b-x\right)f^{\text{'}}\left(x\right)-{\left(b-x\right)}^{2}A$, $\left(A\in R\right)$ and Rolle's theorem is applicable for $\varphi \left(x\right)$ on $\left[a,b\right]$, then which of the following is/are true

Let $y=f\left(x\right)$ be a thrice differentiable function defined on $R$ such that $f\left(x\right)=0$ has at least $5$ distinct zeros, then minimum number of zeros of the equation $f\left(x\right)+6f^{\text{'}}\left(x\right)+12f^{"}\left(x\right)+8f^{\text{'}\text{'}\text{'}}\left(x\right)=0$ is

Prove the following inequality using Lagrange's mean value theorem.

$n\ln \left(1+\frac{1}{n}\right)\le 1 \forall n\ge 1$

Prove the following inequality using Lagrange's mean value theorem.

$\frac{1}{28}<{\left(28\right)}^{\frac{1}{3}}-3<\frac{1}{27}$

Prove the following inequality using Lagrange's mean value theorem.

$\left|\cos a-\cos b\right|\le \left|a-b\right|$

Prove the following inequality using Lagrange's mean value theorem.

$\sin x<x \forall x\in R^{+}$.

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 $\left[1,2\right]$.

 If $f\left(x\right)=\left(x-p\right)\left(x-q\right)\left(x-r\right)$, where $p<q<r$, are real numbers, then the application of Rolle's theorem on $f$ leads to

Applying mean value theorem on $f\left(x\right)={\log }_{e}x; x\in \left[1,e\right]$ the value of $c=\dots \dots \dots$

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 $\left[-4,5\right]$.