Find the maximum value of the expression $ P = {{x} ^ {2}} - \frac {3} {4x} - \frac {x} {y} $

Question

For positive real numbers x, y satisfy the condition $ x + y \le $1. Find the maximum value of the expression $$ P = {{x} ^ {2}} - \frac {3} {4x} - \frac {x} {y} $$ (Please help not to use derivative because this way I solved. Only use basic inequalities)

Answer 1

Clearly, for $y_0<y_1$ we get

$$x^2- \frac {3} {4x} - \frac {x} {y_0}<x^2- \frac {3} {4x} - \frac {x} {y_1}$$

for any $x$, so this means the maximum will have the largest possible $y$. Since $x+y\leq 1$, this must be $y=1-x$. This means we are searching for the maximum of

$$P=x^2-\frac{3}{4x}-\frac{x}{1-x}$$

where $0<x<1$. We will first rewrite it:

$$P=x^2-\frac{4x^2-3x+3}{4x(1-x)}$$

Now since $4x(1-x)$ is symmetric over $\tfrac12$, we will substitute $z=x-\tfrac12$ and we get

$$P=(\tfrac12+z)^2-\frac{4(\tfrac12+z)^2-3(\tfrac12+z)+3}{4(\tfrac12+z)(\tfrac12-z)}$$

which simplifies to

$$P=(\tfrac12+z)^2-\frac{4z^2+z+\tfrac52}{1-4z^2}$$

and now we'll write the whole thing as one fraction:

$$P=-\frac{4z^4+4z^3+4z^2+\tfrac{9}4}{1-4z^2}$$

and from this, we can see it is useful to write it as

$$P=-4z^2\frac{z^2+z+1}{1-4z^2}-\frac{\tfrac94}{1-4z^2}$$

Now simply note that $z^2+z+1>0$ on $|z|<\tfrac12$, so that $\frac{z^2+z+1}{1-4z^2}$ is positive; together with $-4z^2\leq 0$, this means

$$-4z^2\frac{z^2+z+1}{1-4z^2}\leq 0$$

where equality is being reached at $z=0$. Luckily, we also see the maximum of the other term $\frac{9/4}{1-4z^2}$ has a maximum at $z=0$ too (since $1-4z^2\leq 1$ so that $-\tfrac{9/4}{1-4z^2}\leq -\tfrac94$). This means the maximum of $P$ is being found at $z=0$ or $x=\tfrac12$, and $P=-\tfrac94$.

Answer 2

$$P(x,y)-P\left(\frac{1}{2},\frac{1}{2}\right)=-\frac{(1-x-y)x}{y(1-x)}-\frac{(2x-1)^2\left(x^2+3\right)}{4x(1-x)}\leq 0$$