Find the smallest and largest points of intersection for f(x)=x^5 and g(x)=5^x.

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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 **)

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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

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