Question

Compare the growth rate of the functions f(x) = 4x and g(x) = the square root of the quantity of 16 raised to the x power plus 2 raised to the x power.

a. f(x) grows faster than g(x).
b. f(x) and g(x) grow at the same rate.
c. g(x) grows faster than f(x).
d. It cannot be determined.

Answer 1

exponentials eventually grow faster than any polynomials
16^x + 2^x grows faster than 16x^2
g = 16^x + 2^x = 2^(4x) + 2^x
g' = ln2 * 5*2^x
let F(x) = (4x)^2
F' = 32x
so, as soon as ln32 * 2^x > 32x g outstrips F
x > 8.32