Part A: Find the absolute minimum and absolute maximum values of f on the given
ID: 2874122 • Letter: P
Question
Part A:
Find the absolute minimum and absolute maximum values of f on the given interval.
f(x) = x ? ln 9x, [1/2,2]
Part B:
Find the absolute maximum and absolute minimum values of f on the given interval.
f(t) = 2 cos t + sin 2t, [0, ?/2]
Part C:
After an antibiotic tablet is taken, the concentration of the antibiotic in the bloodstream is modeled by the function
C(t) = 8(e?0.4t ? e?0.6t)
where the time t is measured in hours and C is measured in µg/mL. What is the maximum concentration of the antibiotic during the first 12 hours? (Round your answer to four decimal places.)
Part C:
After an antibiotic tablet is taken, the concentration of the antibiotic in the bloodstream is modeled by the function
C(t) = 8(e?0.4t ? e?0.6t)
where the time t is measured in hours and C is measured in µg/mL. What is the maximum concentration of the antibiotic during the first 12 hours? (Round your answer to four decimal places.)
Explanation / Answer
A.
f(x) = x - ln(9x) , x E [1/2 , 2]
We need to find the absolute maximum and absolute minimum value of the given function.
Lets first differentiate the given function
=> f '(x) = 1 - 1/(9x)*9 = 1 - 1/x
Next we find the critical point
=> f '(x) = 0
=> 1 - 1/x = 0
=> x = 1 ,---> This is the critical point
Next we to find the value of the function f(x) at the critical point and at the end points of the
given interval.
Doing so would lead us to our answer.
=> f(1) = 1 - ln(9*1) = 1 - ln(9) = -1.1972
f(1/2) = 1/2 - ln(9*1/2) = 1/2 - ln(4.5) = -1.0041
and f(2) = 2 - ln(9*2) = 2 - ln(18) = -0.89037
Hence from the above result we could see that
The absolute maximum value is f(2) = 2 - ln(18) and is attained at x = 2
The absolute minimum value is f(1) = 1 - ln(9) and is attained at x = 1