What are the extrema of #f(x) = 5 + 9x^2 − 6x^3#?

1 Answer
Dec 15, 2015


Max at #x = 1# and Min #x=0#


Take the derivative of the original function:
#f'(x) = 18x-18x^2#
Set it equal to 0 in order to find where the derivative function will change from a positive to a negative, this will tell us when the original function will have its slope change from positive to negative.
#0 = 18x-18x^2#
Factor a #18x# from the equation
#0 = 18x(1-x)#
#x = 0,1#

Create a line and plot the values #0# and #1#
Enter the values before 0, after 0, before 1, and after 1
Then indicate what parts of the line plot are positive and which are negative.
If the plot goes from negative to positive (low point to a high point) it is a Min if it goes from positive to negative (high to low) it is a max.
All values before 0 in the derivative function are negative. After 0 they are positive, after 1 they are negative.
So this graph is going from low to high to low which is 1 low point at 0 and 1 high point at 1