LaGrange's Mean Value theorem

Q.

Let $\mathrm{f}:\mathrm{R}\to \mathrm{R}$ be a positive increasing function with $\underset{\mathrm{x}\to \infty }{\lim }\frac{\mathrm{f}\left(3\mathrm{x}\right)}{\mathrm{f}\left(\mathrm{x}\right)}1.$ Then, $\underset{\mathrm{x}\to \infty }{\lim }\frac{\mathrm{f}\left(2\mathrm{x}\right)}{\mathrm{f}\left(\mathrm{x}\right)}=1$ is equal to

1. $1$

2. $\frac{2}{3}$

3. $\frac{3}{2}$

4. $3$

Q.

By LMVT, which of the following is true for x>1

1. 1+x In x<x<1+In x
   

2. 1+In x<x<1+x In x

3. x <1 + x In x<1+In x <x

4. 1+In x <1+x In x<x

Q. Let $f:[0,2]\to R$ be continuous on $[0,2]$ and twice differentiable in $(0,2).$ If $f\left(0\right)=0,$ $f\left(1\right)=1$ and $f\left(2\right)=1,$ then

1. $f^{\prime }\left(c\right)=\frac{1}{3}$ for at least one $c\in (0,2)$
2. $2f^{\prime }\left(c\right)+2c=3$ for at least one $c\in (0,1)$
3. $2f^{\prime }\left(c\right)+2c=3$ for at least one $c\in (0,2)$
4. $f^{\prime \prime }\left(c\right)=-1$ for at least one $c\in (0,2)$

Q. If from Lagrange's Mean Value Theorem, we have $f\text{'}\left(x_1\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$, then .

1. $a<x_1\le b$
2. $a\le x_1<b$
3. $a\le x_1\le b$
4. $a<x_1<b$

Q. Let the function $f:[0,8]\to R$ be a differentiable function. Then which of the following is/are correct ?

1. There exist $\alpha ,\beta \in (0,8)$ such that $f^2\left(8\right)=f^2\left(0\right)+16f^{\prime }\left(\alpha \right)f\left(\beta \right);$
2. There exists $\gamma ,\delta \in (0,2)$ such that$\underset{0}{\overset{8}{\int }}f\left(x\right)dx=3\left[{\gamma }^{2}f\right({\gamma }^3)+{\delta }^{2}f({\delta }^3\left)\right];$
3. If $f\left(x\right)=x^5-2x^3-2,$ then it has a real root between $0$ and $2$
4. If $f\left(x\right)=2x^3+3x^2+6x+1,$ then has $3$ real roots.