Find the number of roots of the equation
Now. The exponential function has only 1 value at any point. There is no x value for which the exponential function has more than 1 value.
So, the RHS, inside the brackets, as x varies, there will only ever be 1 unique solution inside the brackets. However outside the brackets we have written so... can you draw a conclusion from there?
so you only need to show that the equation has at most 3 (real) roots. here's a general fact:
Fact: suppose and let be a polynomial of degree with real coefficients. then the equation has at most (distinct) real roots.
Proof: if there's nothing to prove by the fundamental theorem of algebra. for let suppose has at least roots. then, by Rolle's theorem, the
equation must have at least one real root. but contradiction!