Given two real numbers $x,y$ so that $x^{2}+y^{2}+xy+4=4y+3x$. Prove that $3\left(x^{3}-y^{3}\right)+20x^{2}+2xy+5y^{2}+39x\leq 100$.

Question

Given two real numbers $x, y$ so that $x^{2}+ y^{2}+ xy+ 4= 4y+ 3x$. Prove that $$3\left ( x^{3}- y^{3} \right )+ 20x^{2}+ 2xy+ 5y^{2}+ 39x\leq 100$$

I used derivative and Wolfram|Alpha but only the minimum value found $$\min\{3\left ( x^{3}- y^{3} \right )+ 20x^{2}+ 2xy+ 5y^{2}+ 39x\}\Leftrightarrow \left\{\begin{matrix} x\cong 0.0320241\\ y\cong 2.16078\\ \left ( z\cong 2.19235 \right ) \end{matrix}\right.$$ where

• $z$ is a root of $472z^{3}- 449z^{2}- 689z- 1305= 0$,
• $y$ is a root of $27468y- 11800z^{2}+ 17833z- 41733= 0$,
• $x$ is a root of $27468x+ 3304z^{2}- 11167z+ 7722= 0$.

Why is unsuccessful ? Here are two examples of my claim.

Answer 1

From the condition we obtain: $$y^2+(x-4)y+x^2-3x+4=0,$$ which gives $$(x-4)^2-4(x^2-3x+4)\geq0$$ or $$0\leq x\leq\frac{4}{3}.$$ By the similar way we obtain: $$1\leq y\leq\frac{7}{3},$$ which gives: $$3\left ( x^{3}- y^{3} \right )+ 20x^{2}+ 2xy+ 5y^{2}+ 39x\leq$$ $$\leq\frac{64}{9}-3y^3+\frac{320}{9}+\frac{8}{3}y+5y^2+52.$$ Thus, it's enough to prove that: $$\frac{64}{9}-3y^3+\frac{320}{9}+\frac{8}{3}y+5y^2+52\leq100$$ or $$9y^3-15y^2-8y+16\geq0$$ or $$9y^3-24y^2+16y+9y^2-24y+16\geq0$$ or $$(3y-4)^2(y+1)\geq0$$ and we are done!

The equality occurs for $x=y=\frac{4}{3},$ which says that $100$ is a maximal such value.

Answer 2

Alternative solution

$x^2 + y^2 + xy + 4 = 4y + 3x$ is written as $$(y + \tfrac{x}{2} - 2)^2 = \tfrac{1}{4}x(4-3x).\tag{1}$$ From $\frac{1}{4}x(4-3x) \ge 0$, we have $0 \le x \le \frac{4}{3}$.

If $x = 0$, then $y = 2$ and $3(x^3 - y^3) + 20x^2 + 2xy + 5y^2 + 39x = -4 < 100$.

If $0 < x \le \frac{4}{3}$, let $x = \frac{4}{t^2 + 3}$. Then from (1), we have $$y = \frac{2t^2 - 2t + 4}{t^2 + 3}, \quad \mathrm{or}, \quad y = \frac{2t^2 + 2t + 4}{t^2 + 3}.$$ Denote $F = 3(x^3 - y^3) + 20x^2 + 2xy + 5y^2 + 39x$. There are two possible cases:

1. If $y = \frac{2t^2 - 2t + 4}{t^2 + 3}$, we have $$F = \frac{-4t^4+32t^3+128t^2+900}{(t^2+3)^2}$$ and $$100 - F = \frac{8t^2(13t^2-4t+59)}{(t^2+3)^2} \ge 0.$$

2. If $y = \frac{2t^2 + 2t + 4}{t^2 + 3}$, we have $$F = \frac{-4t^4-32t^3+128t^2+900}{(t^2+3)^2}$$ and $$100 - F = \frac{8t^2(13t^2 + 4t + 59)}{(t^2+3)^2} \ge 0.$$

We are done.