How do you use the Intermediate value theorem (IVT) to prove there is at least one solution to 2^x = 1/x

Answer 1

Consider the function f(x) = x² - 1/x. This is continuous except at x = 0.

Obvious x=1 is a solution to f(x) = 0.

How do we prove a solution must exist ?

Well f(0.5) = 0.5² - (1/0.5) = -1.75 < 0

f(2) = 2² - (1/2) = 3.5 > 0

So f(0.5) < 0, and f(2) > 0 and f(x) is continuous on (0.5, 2)

So by the IVT there must be at least one solution of f(x) = 0, on (0.5, 2)

[ imagine stretching a string from (0.5, -1.75) to (2, 3.5) - it must cross the x axis at least once ]

That proves your result

Mike