# 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.**

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 as it is the product of two continuous functions.

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

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

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