<s>[INST] Minimise $x+y$ under the condition $2x+32y \leq 9xy$ where $x,y > 0$ Let $x$ and $y$ be positive real numbers such that $2x+32y\leq 9xy$. Then find the smallest possible value of $x+y$. I tried using AM-GM but it is difficult to find the case of equality. Is Cauchy-Schwarz also helpful? Thanks. 
 [/INST] here is how i approached. i will look at the boundary of $$2x + 32 y = 9xy  $$  which is given by the hyperbola $$ y = \frac{2x}{9x - 32}, \frac{dy}{dx} = -\frac{64}{(9x-32)^2} $$ this hyperbola has slope of $-1$ at $$ -\frac{64}{(9x-32)^2} = -1 \to \left(x = \frac{40}9,y = \frac{10}9, x + y = \frac{50} 9\right) \text{ and} \left(x = \frac{24}9,y = -\frac{6}9, x + y = 2\right) $$ the second solution must be rejected because of the condition that $x > 0$ and $y > 0.$
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