Find the maximum value of $y\cdot{x^2}+x$ if $y^2+x^2+x+y=100$.

Question

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?

Answer 1

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 $$h(x)=\left(-1/2\pm\sqrt{\frac{1}{4}+100-x^2-x}\right)x^2+x$$