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

Find all the local maxima, local minima, and saddle points of the function. f(x,y) = ?(56x^2 - 4y^2 - 16x - 31) + 1 - 8x Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. A local maximum occurs at (Type an ordered pair. Use a comma to separate answers as needed.) The local maximum value(s) is/are (Type an exact answer. Use a comma to separate answers as needed.) B. There are no local maxima.

Transcript

00:01 Given f xy is equal to under the root 56 x square minus 4y square minus 16 x minus 31 plus 1 minus 8 x so now we first find fx so this comes out to be equal to 112 x minus 16 by 2 under the root 56 x square minus 4 y square minus 16 x minus 16 minus 30 minus 8 and we find fy also so this comes out to be equal to minus 8 y by 2 under the root 56 x square minus 4 y square minus 16 x minus 31 now we put fx equal to 0 put fx equal to 0 so from here we get 112 x minus 16 is equal to 16 under the root 56 x square minus 4 y square minus 4 y square minus 16 x minus 31 so simplifying this 7x minus 1 is equal to under the root 56 x square minus 4y square minus 16 x minus 31 so from squaring both sides and simplifying this we can get x plus 2 7x minus 16 is equal to 4 y square now we put fy equal to 0 so on putting fy equal to 0 we will simply get f i equal to 0 we get y is equal to 0 so therefore from this equation we can get that x plus 2 and 7x minus 16 equal to 0 so from here we get x is equal to minus 2 and x is equal to 16 by 7 so critical points in this case become minus 2 0 and 16 by 7 0 next we find f x x x x will come out to be equal to minus 8 28 28 y square plus 225 divided by 56x square minus 16x minus 4y square minus 31 to the power 3 by 2 and fxx at minus 2 0 is equal to minus 8 by 15 next we find f x x at minus f x x at point 16 by 70 which is also equal to minus 8 by 15 so next we find f xy.

02:39 So f xy comes out to be equal to 32y 7x minus 1 by 56 x square minus 4y square minus 16x minus 31 to the power 3 by 2.

02:52 And f xy at minus 2 0 comes out to be 0 and f xy at 16 by 7 0 comes out to be equal to 0.

03:01 Next we find fyy so fyy will be equal to minus 4, 56x square minus 6.

03:07 16 x minus 31 by 56 x square minus 4y square minus 16 x minus 31 to the power 3 by 2 and we find f y y at minus 2 0 which is minus 4 by 15 which is also equal to f y y at 16 by 7 0 so next we find d so d is equal to f x x into f y y minus f xy whole square...

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