Find all the local maxima, local minima, and saddle points...

Find all the local maxima, local minima, and saddle points of the given function. f(x,y) = x\textsuperscript{2} + xy + y\textsuperscript{2} + 2x - 5y + 2 Select the correct choice below and fill in any answer boxes within your choice. A. There are local maxima located at (Simplify your answers. Type ordered pairs. Use a comma to separate answers as needed.) B. There are no local maxima. Select the correct choice below and fill in any answer boxes within your choice. A. There are local minima located at (Simplify your answers. Type ordered pairs. Use a comma to separate answers as needed.) B. There are no local minima. Select the correct choice below and fill in any answer boxes within your choice. A. There are saddle points located at (Simplify your answers. Type ordered pairs. Use a comma to separate answers as needed.) B. There are no saddle points.

Transcript

00:01 In this question, we are asked to find the local maxima, the local minima, and the subtle points of the function f.

00:06 To do that, we need to solve the system of equations fx equals 0 and fy equals 0 to find the critical points.

00:17 This is a system of equations for finding critical points.

00:20 And then, we will use the critical points to find the absolute extrema and the subtle points.

00:27 In our case, the derivative of f with respect to x equals to 2x minus, 6y and we want it to be equal to 0 the derivative of f with respect to y equals to negative 6x plus 2y plus 16 and we also want it to be equal to 0 all right now this is a system of linear equations the first equation could be divided we could divide we could divide we can divide the first equation by 2 we are going to get x minus 3 y equals to 0 and the second equation could be also divided by 2 we are going to get negative 3x plus y plus 8 equals 0 now we are going to from the first equation x equals to 3y now we are going to plug in x equals to 3y in the second equation we are going to get negative 3 multiplied by 3y plus y now let's move 8 to the right hand side to get negative 8 on the right side negative 3 times 3 y is negative 1 negative 9y negative 9y plus y equals to negative 8y.

01:48 So in the second equation, we are going to get negative 8 y equals to negative 8.

01:54 And in the first equation, x equals to 3y.

02:00 Then y equals to 1.

02:02 And after plugging in y equals 1 in the first equation, we are going to get that x equals 3.

02:07 So this is the only critical point of our function.

02:16 Now we will use the second derivative test.

02:26 By the second derivative test, we need to calculate the quantity called d of xy.

02:31 Which equals to f double x multiplied by f double y minus f x y squared.

02:40 Where f double x is the second derivative of f with respect to x, recall that the first derivative equals to 2x minus 6y.

02:50 And if we differentiated one more time with respect to x, we are going to get 2.

02:56 F double y equals to the second derivative of f with respect to y, recall that the first derivative equals to negative 6x plus 2y plus 16.

03:06 So if we differentiate it one more time with respect to y, we are going to get 2.

03:13 And fxy is the derivative of fx with respect to y...

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