Mean Value Theorem

The value of $\text{'}c\text{'},$ of the Lagrange's mean value theorem, for $f\left(x\right)=\sqrt{x^2-x}, x\in [1,4]$ is

If the function $f\left(x\right)=a{x}^3+b{x}^2+26\mathrm{x}-24$ satisfies the conditions of Rolle's theorem in $[2, 4]$ and $f^{\text{'}}\left(3+\frac{1}{\sqrt{3}}\right)=0,$ then the value of $ab=$

If $f\left(x\right)=x^{\alpha }\log x$ and $f\left(0\right)=0,$ then the value of $\text{'}\alpha \text{'}$ for which Rolle's theorem can be applied in $[0, 1]$ is

 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 $(3, 27).$

The point on the curve $y=x(x-4), x\in \left[0,4\right]$, where the tangent is parallel to the $x-\mathrm{axis}$ is

Let $f\left(x\right)=\sin x+\left(x^3-3x^2+4x-2\right)\cos x$ for $x\in (0,1).$ Consider the following statements
I. $f$ has a zero in $\left(0, 1\right)$
II. $f$ is monotone in $\left(0, 1\right)$
Then

Let $R$ be the set of all real numbers and $f\left(x\right)={\sin }^{10}x\left({\cos }^{8}x+{\cos }^{4}x+{\cos }^{2}x+1\right)$ for $x\in R.$ Let $\mathrm{S}=\{\lambda \in R |$ there exits a point $c\in (0,2\pi )$ with $\left.f^{\text{'}}\left(c\right)=\lambda f\left(c\right)\right\}$
Then

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