# Thread: When is the concentration greatest?

1. ## When is the concentration greatest?

The concentration C of insulin in the bloodstream t hours after injection into muscle tissue is

C(t)=4t/2+t^3

When is the concentration greatest? t=?

2. take the derivative of C w/r to time, set C'(t) = 0 and solve for critical value(s) of t, then use the 1st derivative test to check which critical value(s) yields a minimum.

3. Originally Posted by skeeter
take the derivative of C w/r to time, set C'(t) = 0 and solve for critical value(s) of t, then use the 1st derivative test to check which critical value(s) yields a minimum.
Could you help me get the derivative

4. this derivative is fairly basic ... use the quotient rule and show what you get for a derivative.

5. Originally Posted by skeeter
this derivative is fairly basic ... use the quotient rule and show what you get for a derivative.
Is this right? (2+t^3)[4t]-(4t)[2+t^3]
(2+t^3)(4)-(4t)(3t^2)
(8t^3)-(12t^3)
-4t^3

6. Originally Posted by skeeter
What do I do with 8-8t^(3)/ (2+t^3)^2

7. Originally Posted by bgwarhawk
What do I do with 8-8t^(3)/ (2+t^3)^2
*Cough*

Originally Posted by skeeter
take the derivative of C w/r to time, set C'(t) = 0 and solve for critical value(s) of t, then use the 1st derivative test to check which critical value(s) yields a minimum.
So solve $8 - 8t^3 = 0$ etc.

8. Originally Posted by mr fantastic
*Cough*

So solve $8 - 8t^3 = 0$ etc.
How do I get t= if I get t^3=1???

9. Originally Posted by bgwarhawk
How do I get t= if I get t^3=1???
cube root...

10. What do I do with the 1?

11. Originally Posted by bgwarhawk
What do I do with the 1?
Originally Posted by skeeter
take the derivative of C w/r to time, set C'(t) = 0 and solve for critical value(s) of t, then use the 1st derivative test to check which critical value(s) yields a minimum.
t = 1 is the critical value in this case and you are checking if there is a maximum. But the idea is the same.