In mathematics, the mean value theorem states, roughly, that given a planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints.

The Mean Value Theorem states that if f(x) is continuous on [a, b] and differentiable on (a, b) then there exists a number c between a and b such that:

\[\large {f}'(c)=\frac{f(b)-f(a)}{b-a}\]

Solved Example

Question: Evaluate f(x) = x² + 2 in the interval [1, 2] using mean value theorem.

Solution:

Given function is:
f(x) = x² + 2

Interval is [1, 2].

i.e. a = 1, b = 2

Mean value theorem is given by,

f'(c) =

\(\begin{array}{l}\frac{f(b)-f(a)}{b-a}\end{array} \)

f(b) = f(2) = 2² + 2 = 6

f(a) = f(1) = 1² + 2 = 3

So, f'(c) =

\(\begin{array}{l}\frac{6-3}{2-1}\end{array} \)

= 3