The Number of bacteria in a refrigerated food is given by DQ

DQuestion 3 2 pts The number of bacteria in a refrigerated food is given by N (T) = 20712-20T+ 120, for-2 T 14 and where T is the temperature of the food in Celsius. At what temperature will the number of bacteria be minimal? Give the answer as (example): 2 degree Celsius At this temperature, what is the number of bacteria? Give the answer as just the number (example): 5

Solution

N = 20t^2 - 20t + 120

We need N to be minimum..

So, dN/t = 0

dN/dt = 40t - 20 = 0

40t = 20

t = 1/2

Lets confirm that this critical value, t = 1/2
indeed corresponds to the minimum case....

Deriving again :
N\'\' = 40 ---> positive

Since the second derivative > 0,
t = 1/2 corresponds to a point of MINIMUM

So, t = 1/2 -----> ANSWER

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Lets plug in t = 1/2 :
20(1/2)^2 - 20(1/2) + 120

20(1/4) - 10 + 120

115 bacteria