Question:
Find all the zeros of the equation $4^x+6^{x^2}=5^x+5^{x^2}$.
Using the fact 4=5-1, 6=5+1, we can change the equation: $$f(x)=[(5-1)^x-5^x]+[(5+1)^{x^2}-5^{x^2}]=0$$
However... I cannot continue the solution from here. I tried to solve with the derivation of $f(x)$, but nothing I can find.
Actually, I found the zeros using the graphing calculator: $x_1=0$ and $x_2=1$ But, I want to know in a mathematical way. What do you think is the most important key point to the problem? Thanks for your advice.