 # Mean Value Theorem

The Mean Value Theorem is considered to be among the crucial tools in Calculus. According to the theorem, if f(x) is defined and is continuous on the interval [a,b] and can be differentiated on (a,b), then we have at least one value c in the interval (a,b) wherein a<c<b and,

$f’\left ( c \right )=\frac{f\left ( b \right )-f\left ( a \right )}{b-a}$

Rolle’s Theorem is a special case where f(a) = f(b). Here, we have f’(c) = 0. If put differently, there is a point at the interval (a,b) that consists of a horizontal tangent. The Mean Value Theorem can also be stated based on slopes.

$\frac{f\left ( b \right )-f\left ( a \right )}{b-a}$<

The value is a slope of line that passes through (a,f(a)) and (b,f(b)). Therefore, the conclude the Mean Value Theorem, it states that there is a point ‘c’ where the line that is tangential is parallel to the line that passes through (a,f(a)) and (b,f(b)). Example:

Let f(x) = 1/x, a = -1 and b=1.

We know, f(b) – f(a)/b-a

= 2/2 = 1

While, for any cϵ (-1, 1), not equal to zero, we have

f’(c) = -1/c2 ≠ 1

Therefore, the equation f’(c) = f(b) – f(a) / b – a doesn’t have any solution in c. But this does not change the Mean Value Theorem because f(x) is not continuous on [-1,1].