Question

The concentration, in parts per million, of an antibiotic in a human’s bloodstream after x minutes can be modeled by the equation f(x) = 6x 0.02x2 + 5.2 . What is the MAXIMUM amount of the antibiotic in the bloodstream?

Answer 1

If f(x) is 6x/(0.02x^2 +5.2), the maximum concentration of antibiotic is about 9.303 ppm.

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When the numbers are written as rational numbers, the derivative can be found to be
.. f'(x) = 300(260 -x^2)/(260 +x^2)^2
This is zero for
.. x = ±2√65
And the value of f(x) there is
.. f(2√65) = 15√(5/13) ≈ 9.3026