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!

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Answer 1 · 2017-11-10T08:01:52

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

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]}