LaGrange's Mean Value theorem

Q.

If f (x) is differentiable in the interval [2, 5], where $f\left(2\right)=\frac{1}{5}$ and $f\left(5\right)=\frac{1}{2}$, then there exists a number c, 2 < c < 5 for which f ' (c) is equal to

1. 1/5

2. 1/10

3. None of these

4. 1/2

Q.

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

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

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

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

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

Q.

According to LMVT, if a function f(x) is continuous on [a, b] and differentiable on the interval (a, b) then which of the following option should be correct for some value c from the interval (a, b)?( c can take any value from the interval (a, b) )

1. None of the above

2. 

3. 

4. 

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

1. $2f^{\prime }\left(c\right)=g^{\prime }\left(c\right)$
2. $f^{\prime }\left(c\right)=g^{\prime }\left(c\right)$
3. $f^{\prime }\left(c\right)=2g^{\prime }\left(c\right)$
4. $2f^{\prime }\left(c\right)=3g^{\prime }\left(c\right)$

Q. Let f be differentiable for all x. If $f\left(1\right)=-2$ and $f\left(x\right)\ge 2$ for all $x\in \left(1,6\right]$, then which of the following cannot be the value of f(6)?

1. 9
2. 10
3. 6
4. 7
5. 8

Q. For the function $f\left(x\right)=x+\frac{1}{x},x\in \left(1,3\right]$, the value of c for which the Lagranges Mean Value Theorem holds is .

1. 1
2. $\sqrt{3}$
3. 2
4. $\sqrt{2}$

Q. If from Lagranges Mean Value Theorem, we have $f\left(x_1\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$, 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 $f^{\prime }\left(x\right)=e^{x^2}$ and $f\left(0\right)=10.$ If $A<f\left(1\right)<B$ can be concluded from the mean value theorem, then the largest value of $(A-B)$ equals

1. $e$
2. $1-e$
3. $e-1$
4. $1+e$

Q. The value of $c$ in the Lagrange's mean value theorem for the function $f\left(x\right)=x^3-4x^2+8x+11$, where $x\in [0,1]$ is :

1. $\frac{4-\sqrt{7}}{3}$
2. $\frac{2}{3}$
3. $\frac{\sqrt{7}-2}{3}$
4. $\frac{4-\sqrt{5}}{3}$

Q. Let the function, $f:[-7,0]\to R$ be continuous on $[-7,0]$ and differentiable on $(-7,0)$. If $f(-7)=-3$ and $f^{\prime }\left(x\right)\le 2,$ for all $x\in (-7,0),$ then for all such functions $f,f(-1)+f\left(0\right)$ lies in the interval:

1. $[-6,20]$
2. $(-\infty ,20]$
3. $(-\infty ,11]$
4. $[-3,11]$