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.
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.