The IVT

Let f(x) = 1/2^x + 1/3^x. Show that there is a unique, positive number x_0 such that f(x_0)=1.

The Title of this question is “IVT”, which means we will have to use the I.V.T. instead of finding the x value directly. Let’s look at what the IVT says:

"Suppose f is a continuous function on [a,b] and y is between f(a) and f(b). Then there is a number c between a and b such that f(c)=y."

Note f(0)=\frac{1}{2^0}+\frac{1}{3^0}=1+1=2
and f(2)=\frac{1}{2^2}+\frac{1}{3^2}=\frac{1}{4}+\frac{1}{9}=\frac{13}{36}

Since we have not spoken of how to prove whether or not a function is continuous, we are just going to assume that f(x) is indeed continuous on [0,2].

Note that 1 is between f(0) and f(2).

Thus, by the IVT, there exists a c \in (0,2) (c is positive) such that f(c)=1.

I do not know how to prove uniqueness of c, that I leave to someone else.