Parthasarathi Mukhopadhyay and Manabendra Nath Mukherjee Solutions for Exercise 2: EXERCISE

Verify Lagrange's Mean Value Theorem, whenever applicable for the following function:

$f\left(x\right)=x(x-4)$ in $[1, 5]$

Verify Lagrange's Mean Value Theorem, whenever applicable for the following functions:

$f\left(x\right)=2x^2-7x+10$ in $[2, 5]$

Verify Lagrange's Mean Value Theorem, whenever applicable for the following function:

$f\left(x\right)=x^p$ in $[-2, 1]$, where $0<p<1$

Verify Lagrange's Mean Value Theorem, whenever applicable for the following function:

$f\left(x\right)=\sqrt{4-x^2}$ in $[2, 4]$

Verify Lagrange's Mean Value Theorem, whenever applicable for the following function:

$f\left(x\right)=x+\frac{1}{x}$ in $[1, 4]$

Verify Lagrange's Mean Value Theorem, whenever applicable for the following function:

$f\left(x\right)=\frac{1}{4x-1}$ in $[1, 4]$

Find the value of $c$ that appears in the conclusion of Lagrange's mean value theorem in $f\left(x\right)=2x^2+3x+4$ in $[1, 2]$. (Write the answer in fraction.)

Find the value of $c$ that appears in the conclusion of Lagrange's mean value theorem in $f\left(x\right)=\sqrt{x}$ in $[4, 9]$. (Write the answer in decimals.)