Using multivariable calculus to prove inequalities and identities. I’ve realized that onevariable calculus can be a great help in proving inequalities and identities, for example if we want to prove that $$(x+1)^2\ge 0, \quad \forall x\in \mathbb R_{\ge0}$$
we can just define a function $f(x)=(x+1)^2=x^2+2x+1$, and we start by $x=0\implies f(0)=1\ge 0$ then we’ll prove that $f’(x)>0$ for $x\ge 0$. Of course this inequality is trivial because any number squared is $\ge 0$. but this is just an example of how we can use calculus to prove some inequalities.
Now if we have a $2$variables inequality like $$(x+y)^2\ge 0,\quad \forall x\in \mathbb R$$
Can we use multivariable calculus to prove it, or more generally Is multivariable calculus able to give us any kind of help in proving multivariable inequalities or identities?

See Lagrange Multipliers.
They are useful for inequalities but usually aren't the nicest way to prove an inequality.
