Thus, we can find the partial derivatives x f and y f using the Chain Rule : 0 , , x c x z y x F 0 x z z F x y y F x x x F 0 0 . 1 . x z z F y F x F z x F F z F x F x z Similar argument can be used to show that z y F F z F y F y z
APPLIED MATHEMATICS/MAT538 HAH/PPM/FSKM/UiTM 17 E18 Find y if xy y x 6 3 3 . E19 Find x z and y z if 1 6 3 3 3 xyz z y x .
APPLIED MATHEMATICS/MAT538 HAH/PPM/FSKM/UiTM 18 Relative Maximums and Minimums Definition A function of two variables y x f , has a local/relative minima at the point b a , if b a f y x f , , for all points y x , in some region around b a , . b a f , is called the local/relative minimum value and b a , is called the critical / stationary point. A function of two variables y x f , has a local/relative maxima at the point b a , if b a f y x f , , for all points y x , in some region around b a , . b a f , is called the local/relative maximum value and b a , is called the critical / stationary point. Note : The relative minima is not the smallest value that the function will ever take. It only says that in some region around the point b a , , the function will always be larger than b a f , . Outside of that region, it is completely possible for the function to be smaller. Likewise, a relative maxima only says that around the point b a , , the function will always be smaller b a f , . Again, outside of that region, it is completely possible for the function to be larger. Theorem : If y x f , has a local maximum or minimum at b a , and the first order partial derivatives of y x f , exist there, then 0 , b a f x and 0 , b a f y .
APPLIED MATHEMATICS/MAT538 HAH/PPM/FSKM/UiTM 19 Second Derivatives Test Suppose the second partial derivative of f are continuous on a disk with centre b a , , and suppose that 0 , b a f x and 0 , b a f y (that is b a , is a critical point of f ). Let 2 , , , , b a f b a f b a f f f f f b a D D xy yy xx yy yx xy xx If 0 D and 0 , b a f xx then b a f , is a local minimum If 0 D and 0 , b a f xx then b a f , is a local maximum If 0 D then b a f , is not a local minimum or maximum and is called the saddle point If 0 D , the test gives no information. f could have a local minimum or local maximum at b a , or b a , could be a saddle point of f . E20 Find the extreme values of 2 2 , x y y x f . E21 Let 14 6 2 , 2 2 y x y x y x f . Find the local minimum of f . Is this local minimum an absolute minimum?
APPLIED MATHEMATICS/MAT538 HAH/PPM/FSKM/UiTM 20 E22 Find the local maximum and minimum values and the saddle points of 1 4 , 4 4 xy y x y x f
APPLIED MATHEMATICS/MAT538 HAH/PPM/FSKM/UiTM 21 E23 Find and classify the critical points of the functions 4 4 2 2 2 2 4 5 10 , y x y x y x y x f . Find the highest point on the graph of f .
- Fall '16
- Derivative, y , f x, ordered triple x