# Maximum & Minimum of a function

Maximum & minimum Applications Using differentiation techniques to determine maximum values, optimal solutions, of minimum values. Max & Min applications

Skills: Extract relevant information from a word problem, form an equation, differentiate and solve the problem. Class Notes | Blank Notes Graphs: Ex15.04, Maximum & Minimum: Ex15.05, Ex15.06

#### NuLake IAS p89, 90 Guru maths site

# Maximum ##### Positive gradient changing to negative            Mouse over the x values above  ##### find where the derivative = 0 (find x)  # Minimum ##### Negative gradient changing to positive            Mouse over the x values above  ##### To find the minimum, find where the derivative = 0 (find x)  # Point of inflection ##### Gradient decreases to 0 then increases again            Mouse over the x values above  ##### To find this point, find where the derivative = 0 (find x)  ### Example:

 Find the maximum value of the curve y = -x2 - 6x + 8 Steps: y = -x2 - 6x + 8 1) Differentiate to find gradient function y' = -2x - 6 2) The derivative = 0 at a min or max So solve y' = 0 to find the x value for the maximum 0 = -2x - 6 6 = -2x -3 = x 3) Substitute the x value back to find y y = -x2 - 6x + 8 y = -(-3)2 - 6x-3 + 8 y = -9 + 18 + 8 y = 15 Maximum value = 15 Coordinates of the maximum (-3,15)