I need help solving this equation

Question

The temperature T, in degrees Fahrenheit, t hours after 6 AM is given by:

T(t) = − 1 2 t 2 + 8t + 32, 0 ≤ t ≤ 12

What is the warmest temperature of the day? When does this happen?

Answer 1

T(t) -12t^2 + 8t + 32

take the derivative and set equal to zero

T'(t) =-24t + 8 = 0

t = 8/24 = 1/3 hour = 20 minutes

T(1/3) = -12/9 + 8/3 + 32 = 33 1/3

T(0) = 32

T(12) = -12(144) + 8(12) + 32 = -1,600

warmest temperature is 33 1/3 degrees F at 6:20 am

coldest is -1,600 degrees F at 6:00 pm