For which values of x element of the real numbers lays the graph of the function f with function rule f(x) = 2x^4 + 2x^2 below the graph of the function g with function rule g(x) = 5x^3 + 5x? Thank you!

1 Answer
Nov 10, 2017

#f(x)=2x^4+2x^2# lies below the graph of the function #g(x)=5x^3+5x# in the range #0 < x < 5/2#

Explanation:

If the graph of the function #f(x)=2x^4+2x^2# lies below the graph of the function #g(x)=5x^3+5x#,

we must have #g(x) > f(x)#

or #5x^3+5x > 2x^4+2x^2#

or #2x^4+2x^2-5x^3-5x < 0#

or #2x^2(x^2+1)-5x(x^2+1) < 0#

or #(x^2+1)(2x^2-5x) <0#

but #x^2+1>0#

hence we should have #2x^2-5x<0#

or #x(2x-5)<0# i.e. #x(2x-5)# is negative.

This will be

either when #x<0# and #2x-5>0# i.e. #x<0# and #x>5/2#, which is just not possible

or when #x>0# and #2x-5<0# i.e. #x>0# and #x<5/2#, which is possible when #0 < x < 5/2# i.e. when #x# lies between #0# and #2.5#.

That is #f(x)=2x^4+2x^2# lies below the graph of the function #g(x)=5x^3+5x# in the range #0 < x < 5/2#

This is apparent from the following graph of two functions (not drawn to scale i.e. compressed along #y#-axis).

graph{(5x^3+5x-y)(2x^4+2x^2-y)=0 [-4, 5, -50, 200]}