Consider the function z = x^9y + 18x^8

Consider the function z = x^9y + 18x^8 - 512y. Find and classify all critical points of the function. If there are more blanks than critical points, leave the remaining entries blank. The critical point with the smallest x-coordinate is ( , ) Classification: (local minimum, local maximum, saddle point, cannot be determined) The critical point with the next smallest x-coordinate is ( , ) Classification: (local minimum, local maximum, saddle point, cannot be determined) The critical point with the next smallest x-coordinate is ( , ) Classification: (local minimum, local maximum, saddle point, cannot be determined)

Transcript

00:01 We're going to find the critical points of this function.

00:03 The critical points are where the partial derivative of z with respect to x is equal to zero and the partial derivative of z with respect to y is equal to zero.

00:16 Okay so the partial derivative of z with respect to x this is just partial derivative z with respect to x.

00:24 Okay that's another way of of writing it.

00:27 Okay it's equal to we differentiate this function with respect to x while holding y constant.

00:34 Okay so we get 9x to the 8th power times y plus 8 times 18 times x to the 7th power.

00:43 And now let's find the partial derivative of z with respect to y.

00:49 Okay so we'll differentiate with respect to y while holding x constant.

00:52 So we get x to the 9th minus 512.

00:58 And notice that we can write this as x cubed cubed.

01:04 Okay because when you raise a power to a power you just multiply them and we get 9 minus 512 is 8 cubed.

01:16 Okay so i can write that as 8 cubed.

01:19 So then to find the critical points i set each of these equal to zero and i solve for x and y.

01:24 I'm going to start with this equation.

01:28 This is called this expression is called a difference of cubes and there's a formula for finding the factors of it.

01:36 So the formula is a cubed minus b cubed is equal to a minus b times a squared plus ab plus b squared.

01:50 Okay so in our case a is equal to x cubed and b is equal to 8.

02:00 So let's use that formula to factor out x to the 9th minus 512.

02:07 So we get a minus b so x cubed minus 8 times x cubed squared which is x to the 6th power plus 8 x cubed plus 8 squared which is 64 and it's equal to zero.

02:29 Now notice that x cubed minus 8 is also a difference of cubes.

02:33 Okay it's x cubed minus 2 cubed.

02:38 Okay so let's factor that out as well.

02:42 So now a is equal to x and b is equal to 2.

02:50 So we have x minus 2 times x squared plus 2x plus 4 and then we still have x to the 6th plus 8x cubed plus 64.

03:07 Okay so looking at each factor our solutions are x equals 2...

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