(1 point) Suppose f(x, y) = xy(1 - 9x

(1 point) Suppose f(x, y) = xy(1 - 9x - 8y). f(x, y) has 4 critical points. List them in increasing lexographic order. By that we mean that (x, y) comes before (z, w) if x < z or if x = z and y < w. Also, determine whether the critical point a local maximum, a local minimim, or a saddle point. First point ( 0 , 0 ). Classification: saddle point (local minimum, local maximum, saddle point, cannot be determined). Second point ( 0 , 1/8 ). Classification: saddle point (local minimum, local maximum, saddle point, cannot be determined). Third point ( 1/27 , 1/24 ). Classification: local maximum (local minimum, local maximum, saddle point, cannot be determined). Fourth point ( 1/18 , 1/27 ). Classification: saddle point (local minimum, local maximum, saddle point, cannot be determined). Note: You can earn partial credit on this problem.

Transcript

00:01 In this problem, we are provided with the function f of x comma y equals to x times y times 1 minus 9x minus 8 y.

00:15 And we are asked to find out the local extrema of this function.

00:21 So here we can rewrite this function as x times y minus 9x squared y minus 8 x y minus 8 x y squared.

00:31 So now we need to first find out the critical points.

00:35 For that we need the first partial derivatives.

00:38 Partially differentiating with respect to x, we have y minus 18 times xy minus 8 times y squared.

00:47 Equating this first derivative to 0, we have y times 1 minus 18 x minus 8 y equals to 0, which implies that y equals to 0.

01:01 To 0 or 18x plus 8y equals to 1.

01:07 Let us mark this as equation number 1.

01:09 Next we partially differentiate with respect to y.

01:12 We obtain x minus 9x squared minus 16 x.

01:18 So here again equating this first derivative to 0, we have x times 1 minus 9x minus 16 y equals to 0, which implies that either x equals to 0 or 9x plus 16 y equals to 1 let us mark this as equation number 2 so let us multiply 2 with equation number 2 we obtain 18x plus 32y equals to 2 and now let us subtract this with equation number 1 that is with equation 18x plus 8y equals to 1 subtracting both of these, we obtain 24 times y equals to 1, which gives y equals to 1 over 24.

02:11 So, substituting this in the first equation, we have 18 times x plus 1 over 3 equals to 1.

02:23 Solving 4x, we obtain the value of x to be 1 over 27.

02:27 So now we see that we have obtained a critical point 1 over 27 comma 1 over 24 and also we have the critical point 0 comma 0.

02:42 Next let us substitute the value of x as 0 in equation 1.

02:48 So substituting x is equal to 0 in equation 1 we obtain the corresponding value of y to be 1 over 8.

02:58 And substituting y is equal to 0 in equation 2 we obtain the corresponding value of x to be 1 over 9.

03:06 So this gives us two more critical points.

03:09 They are 0 1 over 8 and 1 over 9 0.

03:15 So we have obtained 4 critical points.

03:18 So next let us partially differentiate fx again with respect to x...

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