Question

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!

Answer 1

`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]}