# Find the point where the function $C(t) = 0.24t/(t^2 + 5t +4)$ attends its maximum [closed]

The concentration, $C$, in micrograms per deciliter, of a drug in a patient's bloodstream is given by the equation, $C(t) = \dfrac{0.24t}{t^2 + 5t +4}$ , where $t$ is the number of hours after the drug is administered. How long it takes to reach the maximum concentration ?.

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## closed as off-topic by Care Bear, le gâteau au fromage, Ivo Terek, M Turgeon, HaydenAug 5 at 2:04

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You write "algebraically" and tag this calculus, which do you mean? –  mixedmath Aug 4 at 20:39
@mixedmath Your username is so perfect for that question that at first I thought you might be some sort of bot designed to help clarify poorly-tagged questions. –  Kyle Strand Aug 4 at 22:01

This amounts to finding $t$ where $C'(t) = 0$. Hint: use quotient rule for finding the derivative.

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To find at which point a function achieves its maximum you have to find the roots of the derivative and check if at these points the function achieves a local maximum either using the second derivative or the monotonicity of the function.

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You must find the derivative of your function and set it equal to zero:

$$C'(x) = \frac{0.96 - 0.24x^2}{(x^2 + 5x + 4)^2} = 0$$

$$0.24x^2 = 0.96$$

$$x = 2$$

$x = -2$ is another solution, but note that -2 cannot be expressed as a unit of time.

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Sure it can; $-2$ just means "two hours before the drug was administered." What you mean (probably) is that the model is not accurate for any time prior to the administration of the drug. –  Kyle Strand Aug 4 at 22:06

I would have added this as a comment to expand on the previous answer but I don't have enough reputation, apologies

Whilst Varun Iyer's answer of $t = 2$ as the time it takes to reach the maximum concentration is correct - you should also do some form of check to know whether that is a maximum or a minimum (as said by Mary Star) either by inspection of the function itself or by examining the value of the second derivative at this point ($t = -2$ is when the minimum occurs - of course this model doesn't work for negative time (before the drug is administered)

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