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#?

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Answer 1 · 2016-07-02T09:50:34

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

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

#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]}