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

Dec 15, 2015

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

Explanation:

Take the derivative of the original function:
$f ' \left(x\right) = 18 x - 18 {x}^{2}$
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.
$0 = 18 x - 18 {x}^{2}$
Factor a $18 x$ from the equation
$0 = 18 x \left(1 - x\right)$
$x = 0 , 1$

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