Intermediate Value Theorem

If a function is continuous on the closed interval [a, b] and k is any number between f(a) and f(b) then there exists a number, c, within (a, b) such that f(c) = k.

Intermediate Value Theorem

By observing the graph, the intermediate value theorem can be defined another way:

If a function is continuous on the closed interval [a, b], the function takes all values between f(a) and f(b) in this interval.

The intermediate value theorem does not indicate the value or values of c, it only determines their existance.

Example

Prove that the image of 2 exists in function f(x) = x(sen x +1).

The function is continuous in R as it is the product of two continuous functions.

Take the interval Function Interval, and study the value of the extremes:

Extremes

Extremes

Therefore there is a c pertenece Intervals such that f(c) = 2.