Lecture 19 - Partial Derivatives and Extrema of Functions of Two Variables

Transcription

1 Lecture 19 - Partial Derivatives and Extrema of Functions of Two Variables 19.1 Partial Derivatives We wish to maximize functions of two variables. This will involve taking derivatives. Example: Consider the function given by Suppose we were to fix y at 3. Then z = f(x, y) = x 2 y 3 + xy + 4y 2 f(x, 3) = 27x 2 + 3x + 36 As a function of one variable we can take its ordinary derivative to get 54x + 3 Now, without replacing y by a specific number, we can repeat this. Take the derivative of f(x, y), treating y as a constant. We get 2xy 3 + y This is called the partial derivative of f with respect to x. We denote this by f x, z x or f x where is the Greek letter small delta. Similarly, we could take the derivative with respect to y while pretending x is a constant. The result is 3x 2 y 2 + x + 4y 2 This is called the partial derivative of f with respect to y. We denote this by f y, z y or f y When calculating partial derivatives, just keep the following in mind: When calculating f, pretend y is a constant and take the ordinary derivative with x respect to x. When calculating f, pretend x is a constant and take the ordinary derivative with y respect to y. Note that there are definitions of partial derivatives as limits, but we won t need those. It is also extremely important to remember that this is not the same as implicit differentiation! When finding f x you treat y as a constant and not as a function of x.

2 Examples: Find the partial derivatives of the following functions. 1. f(x, y) = x 2 y 2 = (x 2 y 2 ) 1/2 For each of the partial derivatives we use the general power rule: f x = 1 2 (x2 y 2 ) 1/2 (2x), f y = 1 2 (x2 y 2 ) 1/2 ( 2y) 2. f(x, y) = 4x 2 y + x 3 y 3 + y f x = 8xy + 3x 2 y 3 + 0, f y = 4x 2 + 3x 3 y f(x, y) = e 2x+3y f x = e 2x+3y (2), f y = e 2x+3y (3) 4. f(x, y) = e xy 5. f(x, y) = x ln(xy) f x = e xy (y), f y = e xy (x) 1 f x = ln(xy) + x xy y = ln(xy) f y = x xy x = x y The geometric interpretation of partial derivatives is similar to that of the ordinary derivative of a function of one variable. f x represents the slope of the surface pointing in the x direction while f y represents the slope of the surface pointing in the y direction. Second-Order Partial Derivatives Since a function of two variables has two first derivatives, it has four second derivatives. They are f xx = 2 f x 2 f yy = 2 f y 2 f xy = 2 f y x f yx = 2 f x y = take partial derivative with respect to x twice. = take partial derivative with respect to y twice. = take partial derivative with respect to x, then y. = take partial derivative with respect to y, then x. FACT - for any reasonable function, f xy = f yx.

3 Examples: 1. Let f(x, y) = 3xy 2 + 2xy + x 2. Calculate the 4 second derivatives. f x = 3y 2 + 2y + 2x f y = 6xy + 2x And so we have f xx = 2 f yy = 6x f xy = 6y + 2 f yx = 6y + 2 Notice that f xy = f yx as expected. 2. Let f(x, y) = x 2 y 3 + x 4 y + xe y. Calculate the 4 second derivatives. f x = 2xy 3 + 4x 3 y + e y f y = 3x 2 y 2 + x 4 + xe y And so we have f xx = 2y x 2 y f yy = 6x 2 y + xe y f xy = 6xy 2 + 4x 3 + e y f yx = 6xy 2 + 4x 3 + e y Again f xy = f yx.

4 19.2 Extrema of Functions of Two Variables Let f be a function defined on a region containing (x 0, y 0 ). Then f has a local max at (x 0, y 0 ) if there is a circular region R centered at (x 0, y 0 ) such that f(x, y) f(x 0, y 0 ) for every (x, y) in R. This means that there is a region around (x 0, y 0 ) such that the graph of f looks like this Similarly, f has a local min at (x 0, y 0 ) if there is a circular region R centered at (x 0, y 0 ) such that f(x, y) f(x 0, y 0 ) for every (x, y) in R. This means that there is a region around (x 0, y 0 ) such that the graph of f looks like this Local maxes and mins are examples of critical points just as in the one variable case. We say that (x 0, y 0 ) is a critical point of f if f x (x 0, y 0 ) or f y (x 0, y 0 ) does not exist, or f x (x 0, y 0 ) = 0 and f y (x 0, y 0 ) = 0 We will only deal with the second case. So to find critical points of f(x, y), calculate f x and f y, set both equal to 0 and solve.

5 Examples: 1. Find all critical points of f(x, y) = 2x 2 + y 2 + 8x 6y We find the partial derivatives and set them equal to 0 f x = 4x + 8 = 0 f y = 2y 6 = 0 These imply that x = 2 and y = 3. So f has one critical point at ( 2, 3). 2. Find all critical points of f(x, y) = x 2 5y 2 + 8x 10y 13. We find the partial derivatives and set them equal to 0 f x = 2x + 8 = 0 f y = 10y 10 = 0 These imply that x = 4 and y = 1. So f has one critical point at (4, 1). 3. Find all critical points of f(x, y) = x 2 + 6xy + 10y 2 4y + 4 We find the partial derivatives and set them equal to 0 f x = 2x + 6y = 0 f y = 6x + 20y 4 = 0 This one is more difficult because there are x s and y s in both equations. To solve we multiply the first equation by 3 and add the two equations together: 6x 18y = 0 6x + 20y 4 = 0 2y 4 = 0 thus y = 2. If we put this in the first equation we get that 2x + 12 = 0 or x = 6. So f has one critical point at ( 6, 2). 4. Find all critical points of f(x, y) = e (x2 +y 2). We find the partial derivatives and set them equal to 0 f x = e (x2 +y 2) ( 2x) = 0 f y = e (x2 +y 2) ( 2y) = 0 Since e (x2 +y 2) can t be 0, the only critical point we have is (0, 0). Now that we know how to find critical points, we would like to classify them. For our purposes, if (x 0, y 0 ) is a critical point but neither a local max nor a local min we will say that it is a saddle point. Then we have the following theorem

6 Theorem 19.1 Suppose f(x, y) is a function and that (a, b) is a critical point of f. Define D = f xx (a, b)f yy (a, b) [f xy (a, b)] 2 Then 1. There is a local max at (a, b) if D > 0 and f xx (a, b) < There is a local min at (a, b) if D > 0 and f xx (a, b) > There is a saddle point at (a, b) if D < If D = 0, WHO KNOWS!?!?! Example: Recall our first example f(x, y) = 2x 2 + y 2 + 8x 6y We had one critical point at ( 2, 3). f x = 4x + 8 So f y = 2y 6 f xx = 4 f yy = 2 f xy = 0 So D = 4(2) 0 = 8 > 0 and f xx > 0. Thus ( 2, 3) is a local min.