LaGrange's Mean Value theorem

Q.

What is Cauchys mean value theorem?

Q.

Find the domain and range of the following real function:

(i) f(x) = –|x| (ii)

Q.

In $\left[0,1\right]$ Lagrange’s mean value theorem is NOT applicable to

1. $f\left(x\right)=\left\{\begin{array}{l}\left(\frac{1}{2}-x\right)x<\frac{1}{2}\\ {\left(\frac{1}{2}-x\right)}^{2}x\ge \frac{1}{2}\end{array}\right.$

2. $f\left(x\right)=\left\{\begin{array}{l}\frac{\sin x}{x}x\ne 0\\ 1x=0\end{array}\right.$

3. $f\left(x\right)=x\left|x\right|$

4. $f\left(x\right)=\left|x\right|$

Q. Let $A$ and $B$ be two sets defined as $A=\{x:x\in W\text{and}-1\le \frac{2x+3}{5}\le 3\}$ and $B=\{x:x\in Z\text{and}0\le \frac{3-x}{7}\le 1\}$. If $P=A-B$ and $Q=B-A$, then the value of $n(P\times Q)$ is

1. $12$
2. $15$
3. $24$
4. $16$

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.

The value of $c$ in mean value theorem for the function $f\left(x\right)=2x^2+3x+4$ in the interval $\left[1,2\right]$ is

1. $\frac{1}{2}$

2. $\frac{1}{3}$

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

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

Q.

Let f be a function defined on [a, b] such that f'(x) > 0, for all $xϵ$ [a, b]. Then prove that f is an increasing function on [a, b].

Q.

A function $f\left(x\right)=1+\left(\frac{1}{x}\right)$ is defined on the closed interval $[1,3]$. A point in the interval, where the function satisfies the mean value theorem, is

1. $x=\frac{1}{3}$

2. $x=\frac{1}{\sqrt{3}}$

3. $x=\sqrt{3}$

4. None of these

Q.

Verify Lagranges Mean Value theorem for the given function: $f\left(x\right)=2x^2-7x+10$ in $[2,5]$.

Q. If the function $f\left(x\right)=3x^4-8x^3+12x^2-48x+25$ has global maximum value $M$ and global minimum value $m$ in $[0,3],$ then $2M+m$ is equal to

Q.

What is Cauchy Mean Value Theorem?

Q.

The sum and sum of squares corresponding to length x (in cm) and weight y

(in gm) of 50 plant products are given below:

Which is more varying, the length or weight?

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.

Show by using mean value theorem and taking f(x)=logx that 1-a/b < logb/a < b/a -1

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.

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. Let $p\left(x\right)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p\left(1\right)=6\text{and}p\left(3\right)=2$, then $p^{\prime }\left(0\right)$ is

Q.

If the mean value theorem is $f\left(b\right)-f\left(a\right)=(b-a)f\left(c\right)$. Then for the function $x^2-2x+3$ in $\left[1,\frac{3}{2}\right]$, the value of $c$ is

1. $\frac{6}{5}$

2. $\frac{5}{4}$

3. $\frac{4}{3}$

4. $\frac{7}{6}$

Q. State 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. If $f:[-5,5]\to R$ is a differentiable function and if $f^{\prime }\left(x\right)$ does not vanish anywhere, prove that $f(-5)\ne f\left(5\right)$.

Q.

Mean Value Theorem states

1. 

2. 

3. 

4. 

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 is a differentiable function and if does not vanish anywhere, then prove that .

Q.

If $f\left(x\right)=x^3$ and $g\left(x\right)=x^3-4x$ in $-2\le x\le 2$, then consider the statements

I $f\left(x\right)$ and $g\left(x\right)$ satisfy mean value theorem.

II $f\left(x\right)$ and $g\left(x\right)$ both satisfy Rolles theorem.

Which of these statements is true?

1. I and II are correct

2. Only I is correct

3. None is correct

4. Only II is correct

Q.

Verify Mean Value Theorem, if in the interval, where and.

Q.

What is the Mean Value Theorem for integrals?

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. Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2.

Q.

What is the use of the mean value theorem?