Question

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

Answer 1

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

Explanation:

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