The product of two numbers $x$ and $y$ is $16$. We know $x\ge 1$ and $y\ge 1$. What is the greatest possible sum of the two numbers?
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Hint: Since $x \cdot y = 16$ and $x,y \geq 1 > 0$ we have $y = \frac{16}{x}$. Now can you find the maximum of the function
$$f(x) = x + \frac{16}{x}$$
for $x \geq 1$ and $\frac{16}{x} \geq 1$?
Surb
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I'd suggest the graphical method. Draw $xy=16$, draw $x=1$ and $y=1$, draw $x+y=k$ for some $k$. Behold! – Joker_vD Sep 15 '14 at 15:04
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We want to maximise $f(x)=x+\frac{16}{x}$ over $1\le x\le 16$. $f'(x)=0$ at precisely $x=4$. Now compare the values of $f$ at the points $1,4,16$ to find the extrema.
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