Use the formula for the curve that is provided or that you derive based on your data. For example, you are looking at a graph of the population of bacteria and want to know the maximum number of cells that existed. You find that the graph is equal to - x^2 + 6x - 4.
Take the first derivative of the formula and solve for x.
f(x) = -x^2 + 6x - 4.
f ' (x) = -2x + 6
6 = 2x
x = 3
Replace x in the original equation and solve for the extreme value.
f(3) = - (3)^2 + 6(3) - 4
f(3) = 5
Take the second derivative of the equation to determine if the value is a maximum or minimum. If the second derivative is positive then it is a minimum, otherwise it is a maximum.
f' (x) = -2x - 6
f" (x) = - 2
We now know that the value is a maximum because f" (x) = - 2.
Use the two data points for the extreme value found in the equations above. The x value was solved after taking the first derivative and the y value found by plugging x back into the original equation. The maximum point is at (3,5).