NOTES: Lab 3, ClairautHere are some notes to help you with the algebra in the lab:1. Letf(x, y) =xy(x2− y2)x2+ y2Notice that this could be written as:f(x, y) = xy · x2x2+ y2−y2x2+ y2!To show that −|xy| ≤ f(x, y) ≤ |xy|, show that the quantity in paren-thesis is between ±1:We note that the two numbers in parenthesis must sum to 1. We couldrephrase that: Let a + b = 1, where 0 ≤ a, b ≤ 1. Find the max andmin of a −b. We could do a substitution, so we would find the max andmin of a − (1 −a) = 2a − 1, where a is between 0 and 1. The maximumis attained at a = 1 (b = 0) and the minimum is where a = 0 (b = 1).Therefore,−1 ≤x2− y2x2+ y2≤ 12. Show that fxshould be 0 at (x, y) = (0, 0) by seeing that it is trappedbetween 2|y| and −2|y|. To show this, note that:x4− y4+ 4x2y2(x2+ y2)2=x4+ 2x2y2+ y4+ 2x2y2− 2y4(x2+ y2)2We wrote the fraction in this way to simplify things a bit:x4+ 2x2y2+ y4+ 2x2y2− 2y4(x2+ y2)2=(x2+ y2)2+ 2y2(x2− y2)(x2+ y2)2Now simplify:(x2+ y2)2+ 2y2(x2− y2)(x2+ y2)2= 1 + 2y2x2+ y2 x2x2+ y2−y2x2+ y2!1As before, let a = x2/(x2+ y2), and b = y2/(x2+ y2). Then a, b ≥ 0,a + b = 1, and we want to find the minimum and maximum of:1 + 2b(a − b)Substituting a = 1 − b, we find the min and max of1 + 2b((1 − b) − b) = 1 + 2b(1 − 2b) = 1 + 2b − 4b2, 0 ≤ b ≤ 1Using Calculus, you should find the the maximum occurs at b = 1/4,and the minimum occurs at b = 1. Put these back into the expressionto see that:−1 ≤x4− y4+ 4x2y2(x2+ y2)2≤54Therefore,−|y| ≤ fx(x, y) ≤54|y|Now see if you can do something similar for fy(x, y).3. For the second mixed partials, try plotting. Does the graph look famil-iar (like something from our practice Maple sheet)?Algebraically, take note of fx(0, y) and fy(x, 0). Then compute:fxy(0, 0) = limh→0fx(0, 0 + h) − fx(0, 0)handfyx(0, 0) = limh→0fy(0 + h, 0) − fy(0, 0)hSome things I want you to get from doing Lab 3:• Maple is a very powerful visualization and computational to ol, espe-cially in three dimensions.• We should never trust Maple completely- Always do a “reality check”on what Maple is giving you to see if you believe it.• From our algebra in these notes, it’s clear that while Maple is a greattool, nothing beats old fashioned mathematical reasoning and