How to use Intermediate Value Theorem?

Question

Use the Intermediate Value Theorem to prove that the equation has at least one real solution.

cosx + lnx = x^3 - 4

Answer 1

The Intermediate value theorem states that if a continuous function, y=f(x) crosses the x-axis between two values of x, then f(x) has a zero (or root) between the two values of x.

For the given problem, define the function

f(x) = cos(x) + ln(x) - x² + 4

This function should be zero at a certain value of x. Let us use the Intermediate theorem to verify that it is so.

Let x=1. Then f(1) = cos(1) + ln(1) - 1² + 4 = 3.5403

Let x=2. Then f(2) = cos(2) + ln(2) - 2² + 4 = -3.723

Because the value of the f(x) changes sign between x=1 and x=2, there is a zero between x=1 and x=2.

The graphical calculator verifies that the zero occurs at approximately x = 1.642.