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ight) - 10 = 2(5 - \sqrt{7}) - 10 = -2\sqrt{7} < 0 \quad ( ext{local maximum}).
Thus, the revenue growth rate is maximized at \( t = rac{5 - \sqrt{7}}{9} \).
oxed{ rac{5 - \sqrt{7}}{9}}
A pharmacologist is studying the concentration of a new Alzheimer's drug in the bloodstream, modeled by the function \( C(x) = 5x^2e^{-x} \). Determine the dosage level \( x \) that maximizes the concentration.
To find the dosage level that maximizes the concentration, we first find the derivative of \( C(x) \):
C'(x) = rac{d}{dx}(5x^2e^{-x}).
Using the product rule, where \( u = 5x^2 \) and \( v = e^{-x} \), we have:
C'(x) = u'v + uv' = (10x)e^{-x} + 5x^2(-e^{-x}) = e^{-x}(10x - 5x^2).
Set \( C'(x) = 0 \) to find critical points:
e^{-x}(10x - 5x^2) = 0.