Prove that f(x) = 4x+2^x-2^2x has at least one real zero in the interval [1,2]. Do not find the solution, only show that at least one must exist in the indicated interval

Answer 1

find f(1) and f(2)

plug 1 in place of x in the function. f(1)=2

then plug 2 in place of x in the function. f(2)=-4

one is negative, one is positive.

That means the function must be zero somewhere inbetween because

the graph is continuous. plot the two points (1,2) and (2,-4). You can't connect those two points without crossing the x-axis at least once, somewhere. Where it crosses is the zero.