The concentration C(t) of a certain drug in the bloodstream," t" hours after injection is C(t)=sin⁡t+cos⁡t-1 on the interval [0, π/2]. When does the drug concentration reach its maximum value?"

Answer 1

The question we are trying to answer is when does the concentration reach its maximum, given the concentration function:

C(t) = sin(t) + cos(t) - 1

with t on the interval [0,π/2]. In order to find a maximum or minimum (also known as an extremum) of a function, the standard procedure is to take the derivative and set it equal to zero.

C'(t) = cos(t) - sin(t) = 0,

which implies that

cos(t) = sin(t).

This is only true when t = π/4 on the given interval, so this must be our maximum. In order to determine that this is indeed a maximum and not a minimum, we need to calculate the second derivative and make sure that it is negative at the time t = π/4. The second derivative is given by

C''(t) = - sin(t) - cos(t),

which at t = π/4 gives

C''(π/4) = - sqrt(2)/2 - sqrt(2)/2 < 0.

This shows that the concavity is downward, and therefore the extremum at t = π/4 is indeed a maximum.