LaGrange's Mean Value theorem

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 ^′$f\left(x\right)\le 2$, for all $\in (-7,0)$, then for all such functions, $f(-1)+f\left(0\right)$ lies in the interval:

1. $[-6,20]$

2. $(-\infty ,20]$

3. $(-\infty ,11]$

4. $[-3,11]$

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

Q. Let $f$ be a twice differentiable function defined in $[-3,3]$ such that $f\left(0\right)=-4,f^{\prime }\left(3\right)=0,$ $f^{\prime }(-3)=12$ and $f^{\prime \prime }\left(x\right)\ge -2\forall x\in [-3,3].$ If $g\left(x\right)=\underset{0}{\overset{x}{\int }}f\left(t\right)dt,$ then the maximum value of $g\left(x\right)$ is

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 Lagrange's Mean Value Theorem holds is .

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

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. $[-3,11]$

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. $f^{\prime }\left(C\right)=\frac{f\left(a\right)-a}{f\left(b\right)-b}$

2. $f^{\prime }\left(C\right)=\frac{f\left(a\right)-f\left(b\right)}{a-b}$

3. $f^{\prime }\left(C\right)=\frac{f\left(b\right)-f\left(a\right)}{a-b}$

4. None of the above

Q. Let f be differentiable for all x. If $f\left(1\right)=-2$ and $f\text{'}\left(x\right)\ge 2$ 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.

From mean value theorem $f\left(b\right)-f\left(a\right)=f(b-a)f\left(x_1\right)$; $a<x_1<b$ if $f\left(x\right)=\frac{1}{x}$, then $x_1=$

1. $\sqrt{ab}$

2. $\frac{[a+b]}{2}$

3. $\left[\frac{2ab}{(a+b)}\right]$

4. $\frac{[b–a]}{[b+a]}$

Q. Let $f$ be any function continuous on $[a,b]$ and twice differentaible on $(a,b)$. If for all $x\in (a,b),f^{\prime }\left(x\right)>0$ and $f^{\prime \prime }\left(x\right)<0$, then for any $c\in (a,b),\frac{f\left(c\right)-f\left(a\right)}{f\left(b\right)-f\left(c\right)}$ is greater than :

1. $\frac{b-c}{c-a}$
2. $1$
3. $\frac{c-a}{b-c}$
4. $\frac{b+a}{b-a}$

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. $[-3,11]$