<s>[INST] Proving inequalities using Calculus In general how do you prove inequalities using calculus, I believe it is using maxima or minima right? For example

$$a^2b+b^2c+c^2a \le 3, \qquad a,b,c \ge 0,\quad a+b+c=3.$$

How would you use calculus, just a sketch, I am interested in the method itself. 
 [/INST] $f(a,b) = a^2b + b^2(3-a-b) + (3-a-b)^2a = a^2b + 3b^2-ab^2-b^3+9a+a^3+ab^2-6a^2-6ab+2a^2b=3a^2b+3b^2-b^3+9a+a^3-6a^2-6ab$. Thus: $f_a = 6ab+9+3a^2-12a-6b = 0 = f_b = 3a^2+6b-3b^2-6a\Rightarrow 3(a^2-b^2) = 6(a-b)\Rightarrow (a-b)(a+b-2)=0$. There are $2$ cases:
Case $1$: $a = b \Rightarrow 6a^2+9+3a^2-12a-6a = 0\Rightarrow 9a^2-18a+9 = 0\rightarrow 9(a-1)^2 = 0\to a = 1=b, c = 1$, since $a+b+c=3$.
Case $2$: $a+b = 2 \Rightarrow 6a(2-a)+9+3a^2-12a-6(2-a) = 0\Rightarrow 12a-6a^2+9+3a^2 - 12a-12+6a=0\to -3a^2-3+6a=0\to -3(a-1)^2=0 \to a = 1, b = 1, c = 1$. 
Either case gives $f_{max} = f(1,1,1) = 3\to a^2b+b^2c+c^2a \leq 3$
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