sin(e^x)=5^x+5^-x

Question

find the number of real solutions? options are:- 1. 1 solution 2. 0 solution 3. 2 solution 4. infinitely many solutions

Answer 1

Answer 2: No solutions.

sin(e^x)=5^x+5^-x.

When x≥0 the right side has a minimum value of 2 (when x=0). When x=1, it's 5.2.

When x<0 the right side has the same values since 5^|x|+5^-|x|=5^-|x|+5^|x|. In other words, if x=-1, the right side is 5.2.

The sine function cannot exceed 1 so the left side can never equal the right side.