Use the intermediate value theorem to prove that 2^x+3^x=4^x

Answer 1

Let f(x) = 2^x + 3^x - 4^x. The function is continuous for all x.

f(1) = 1 > 0 and f(2) = -3 < 0

So, by the Intermediate Value Theorem, f(c) = 0 for some number c in the interval (1,2).

2^c + 3^c - 4^c = 0 Therefore, 2^c + 3^c = 4^c.