Find the smallest and largest points of intersection for f(x)=x^5 and g(x)=5^x.
Find the smallest and largest points of intersection for `f(x)=x^5` and `g(x)=5^x` .
(1) As above the smallest point of intersection is `x~~1.7649219` and `y~~17.124878` found using a graphing utility. To compute the exact value you must use the LambertW function -- see reference. (More than likely you were expected to use a graphing utility to find this intersection)
(2) The second point of intersection is the point x=5,y=3125. This is the only rational intersection.
(** exponential functions like `5^x` grow far faster than power functions like `x^5` . In this case, `5^x>x^5` for all x>5 **)
There is only one intersection as the value of x^5 power is negative when x is negative, 0 when x=0, and rises more rapidly then 5^x as x increases above zero. 5^x power is always positive. At x = 2, x^5 is 1 and 5^x is 5. They switch by x=2 to 32 and 25 respectively.
5^x = x^5 is about x = 1.765, y = 17.128666