How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for y=x^4-2x^3?

1 Answer
Jul 2, 2016

intercept / stationary point at x= 0 is an inflexion point

stationary point at x = 3/2 is a relative minimum

Explanation:

y=x^4-2x^3 = x^3(x-2) so y = 0 at x = 0, 2

y' = 4x^3 - 6x^2 = x^2(4x-6) so y' = 0 at x = 0, 3/2

y'' = 12x^2 - 12x = 12x(x-1)

y''(0) = 0 and y''(3/2) = 9 [> 0]

so the intercept and stationary point at x= 0 is an inflexion point

and the stationary point at x = 3/2 is a relative minimum

globally, the dominant term in the expression is x^4 so lim_{x to pm oo} y = + oo

you can see all of this in the plot
graph{x^4 - 2x^3 [-10, 10, -5, 5]}