<s>[INST] Find the maximum value of $y\cdot{x^2}+x$ if $y^2+x^2+x+y=100$. We know that $y^2+x^2+x+y=100$. Find the maximum value of  $$y\cdot x^2+x$$ 
I tried to simplify it and use inequalities but I failed. Is there a way to solve it without calculus?
 [/INST] You Can reduce your Problem in two variables in to one: from $$y^2+y-100+x^2+x=0$$ follows $$y=-\frac{1}{2}\pm\sqrt{\frac{1}{4}+100-x^2-x}$$
so you will get
$$h(x)=\left(-1/2\pm\sqrt{\frac{1}{4}+100-x^2-x}\right)x^2+x$$
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