Mean Value Theorems

If $a,b,c,d\in R$ such that $\frac{a+2c}{b+3d}+\frac{4}{3}=0$, then the equation $a{x}^3+b{x}^2+cx+d=0$ has -

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

If $f\text{ and }\text{g}$ are differentiable functions in $[0, 1]$satisfying $f\left(0\right)=2=\text{g}\left(1\right)\text{, }\text{g}\left(0\right)=0\text{ and }f\left(1\right)=6\text{, }$ then for some $\text{c}\in \left]0\text{, }1\right[$ 

If the Rolle's theorem holds for the function $f\left(x\right)=2x^3+a{x}^2+bx$ in the interval $\left[-1,1\right]$ for the point $c=\frac{1}{2}$, then the value of $2a+b$ is:

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

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 $c$ 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 $k$.

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 

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 $k$.

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 $k$.

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 $(3, 27).$