Lagranges Mean value Theorem

Theorem was first proved by the French mathematician Count Joseph Louis Lagrange (1736-1813). And hence Lagranges mean value theorem

Theorem : Let $f$ be a Continuous function defined on $[a,b]$ and differentiable in $]a,b[$. Then there exists a number $x_o$ in $]a,b[$ such that.
$f'( x_{o}) =\frac{f( b) -f( a)}{b-a}$
Geometrically we can interpret this theorem as given in

In this figure you can see that the straight line connecting the end points $(a,f(a))$ and $(b, f(b))$ of the graph is  parallel to some tangent to the curve at an intermediate point.
You may wonder why this theorem is called ‘mean value theorem’. This is because of   ; the following physical interpretation.
Suppose $f(t)$ denotes the position of an object at time $t$. Then the average (mean) velocity during the interval $[a,b]$ is given by
$\frac{f( b) -f( a)}{b-a}$
Now Theorem states that this mean velocity during an interval $[a, b]$ is equal to ne Velocity $f'( x_{o})$  at some instant $x_o$ in $]a, b[$.
We shall illustrate the theorem with an example.
Example: Apply the mean value theorem to the function $f(x) =\sqrt{x}$ in $[0, 2]$