Expert: Paul Klarreich - 11/19/2009

Question
Please help me to solve this problem. Thank you.

The function: z=(x-2)^4+(y-3)^4. How to find the critical points in the neighborhood of (2,3,0). Why or why not they are/are not local maxima or minima.

Answer
Questioner: Geli
Country: United States
Category: Advanced Math
Private: No
Subject: Maxima or minima
Question: Please help me to solve this problem. Thank you.

The function: z=(x-2)^4+(y-3)^4. How to find the critical points in the neighborhood of (2,3,0). Why or why not they are/are not local maxima or minima.
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Let's start by simplifying:  Let X = x-2,  Y = y-3.  Then (2,3,0) becomes (0,0,0).

Now you get a CP by finding dz/dX and dz/dY

dz/dX = 4X^3.  That is zero at  X = 0.

Further checking of just the graph of  z = X^4 will show that it is a minimum.

dz/dY = 4Y^3.  That is zero at  Y = 0.
Likewise, further checking of just the graph of  z = Y^4 will show that it is also a minimum.

So your point, (2,3,0) is indeed a minimum.