Mean Value Theorems

Let $P$ be a polynomial whose coefficients are real numbers. Suppose the roots of $P\left(x\right)=0$ are real. Then

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 -

 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

A function $f\left(x\right)$ fulfil the conditions of Lagrange's mean value theorem in $\left[0,5\right]$. If $\left|f^{\text{'}}\left(x\right)\right|\le \frac{1}{4} \forall x\in \left[0,5\right]$ & $f\left(0\right)=0$, then which of the following is true

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 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 $(3, 27).$

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

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

If $f\left(x\right)$ be a twice differentiable function such that $f\left(x\right)=x^2$ for $x=1, 2, 3,$ then

Let $a, b$ and $c$ be non-zero real numbers, such that the integral$\underset{0}{\overset{1}{\int }}\left(a{x}^2+bx+c\right)\left(1+{\cos }^{8}x\right)dx=\underset{0}{\overset{2}{\int }}\left(a{x}^2+bx+c\right)\left(1+{\cos }^{8}x\right)dx$ then, the equation $a{x}^2+bx+c=0$ has: