Intermediate Value Theorem

How to use the intermediate value theorem to solve its application problems: theorem, 1 example, and its solution.

Theorem

Theorem

If y = f(x) is continuous in the interval [a, b],
and
if f(a) ≠ f(b),

then x = c exists
that satisfies
f(c) = k
k is between f(a) and f(b).

This is the intermediate value theorem.

Example

Example

Solution

f(x) is a polynomial.
So f(x) is a continuous function.
So f(x) is continuous in the interval [1, 2].

Find f(1) and f(2).

f(1) = -2
So f(1) is minus.

f(2) = 3
So f(2) is plus.

f(1) is minus.
f(2) is plus.

So f(1) ≠ f(2).

f(x) is continuous in [1, 2].
f(1) ≠ f(2)

Then, by the intermediate value theorem,
x = c exists in (1, 2)
that satisfies
f(c) = 0.
This 0 is between f(1), which is (-), and f(2), which is (+).

x = c exists in (1, 2)
that satisfies f(c) = 0.

Then c is the zero of f(x) in (1, 2).

So the zero of f(x) exists in (1, 2).

So, by using the intermediate value theorem,
you can prove that
the zero of f(x) exists in (1, 2).