For what values of x is the graph of y = x4 - 9x both increasing and concave up?.....
how do i figure this out? i graphed the function, and see where it is concave up, but dont know how to tell what the values are...any help?
y = x^4 -9x
y' = 4x^3 -9
y'' = 12x^2
y' determines the slope of the curve. If y' is negative, the graph is going down. If y' is positive, the graph is going up. If y' = 0, the graph is horizontal, neither decreasing nor increasing. So, y' > 0 if graph is increasing.
y' = 4x^3 -9
4x^3 -9 > 0
x^3 > 9/4
x > cubrt(2.25) -----the graph is increasing.
y'' determines the concavity of the curve.
If the y'' is positive, then the concavity of the graph is facing upwards.
y'' = 12x^2
Except for x=0, for any real values of x, the y'' is positive.
Therefore, for the graph of y = x^4 -9x is increasing and concaves up for all values of x, such that x > cubrt(2.25). -----------answer.