Relative Extrema of Two Variable Functions

“Understanding Variables”

Variables in Functions: Function of One Variable… Function of Two Variables… f (x,y) = 4x + 6y - 9 f (x) = 7x - 3

Independent Variables: X (Height) and Y’s (Width) that make up a given equation. z = F (3,2) = 4x + 6y - 9 Independent

z = F (3,2) = 4x + 6y - 9 Dependent Dependant Variables: The Z (Depth), which the equation is trying to solve.

“Understanding Relative Extrema”

Relative Extrema: The highest point in a particular section of graph is called a Local Maximum. The lowest point in a particular section of graph is called a Local Minimum. Maximum Minimum

Relative Extrema: These points can be recognized as the points where the functions derivative equals zero. A derivative is a function, which gives the slope of a line tangent to a function. Maximum Minimum

“Finding Relative Extrema”

Variables and Extrema: As the number of variables in a function changes so does the way one must go about finding the derivatives of those functions. Δ variables = Δ derivative test

Derivatives: One variable functions use the basic derivative test. f (x) = x f’ (x) = 2x Point of Extrema

Partial Derivatives: Two variable functions use the partial derivative test. In this test you must find the derivative of one variable at a time treating the other variable as a real number. f (x,y) = x 2 + 3y - 5 fx = 2x fy = 3 Points of Extrema

“Determining Maximum or Minimum”

Second Derivatives: One variable functions use the basic second derivative test to determine max or min’s. f’ (a) = 0 f’’ (a) > 0 f’’ (a) < 0 If we know that then if or if then this point is a relative maximum then this point is a relative minimum

Second Partial Derivatives: Two variable functions use the second partial derivative test to determine max or min’s. fx (a,b) = 0 and fy (a,b) = 0 fxx (a,b) fyy (a,b) – [fxy (ab)] 2 > 0 fxx (a,b) > 0 fxx (a,b) < 0 fxx (a,b) fyy (a,b) – [fxy (ab)] 2 = 0 or < 0 If we know that then if …and then this point is a relative minimum then this point is a relative maximum but if this point is neither

Everyday Application

Finding Maximum Profit: Say your business requires multiple variables in its profit function, like the one below… …where P represents the profit you make from producing x units of donuts and y units of coffee. P (x,y) = 4x + 6y - 9

Finding the Maximum Profit: Going through the steps outlined in this presentation will help you to find out how many donuts and how many coffees you would need to produce to achieve maximum possible profit. Also from plugging these discovered numbers into the profit equation you can also see what the highest possible profit will be.