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This is a quadratic function that opens downward, so its maximum occurs at the vertex:
Then $ l = 48 - 24 = 24 $, so the rectangle is actually a square. The maximum area is:
Question: A palynologist observes that the concentration of a rare pollen type in a sediment layer is proportional to $
We minimize $ f(t) $. Apply the AM-GM inequality:
Question: An environmental scientist models soil nutrient retention as a function $ f(x) $ satisfying $ f(x + y) = f(x) + f(y) + xy $ for all real numbers $ x, y $. Given that $ f(0) = 0 $, find all such functions $ f $.
This holds, consistent with $ f(0) = 0 $. Now define a new function $ g(x) = f(x) +
ight) + xy + xy = g(x) + g(y) + xy + xy
This is not helpful. Try another substitution. Let:
Substitute into the functional equation:
Equal. So all solutions are of the form: