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The temperature measured during a day in summer can be modeled by the function T(x) = 40 - 6*(x - 6)^2 where x is the number of hours after 6 AM.
This can be represented as a graph of temperature versus time as follows:
The graph is a parabola with temperature rising, reaching a maximum and then dropping.
If the temperature is at the maximum t hours after 6 AM. T'(t) = 0 and T''(t) < 0.
As T(x) = 40 - 6*(x - 6)^2, T'(x) = -12*(x - 6) and T''(x) = -12.
T'(t) = 0
=> t = 6
And T''(6) is negative.
The temperature is at the maximum at 12 PM.
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