A biologist modelled the population, \(N\), of bacteria in a sample by the equation \(N=-t^3+\dfrac{41}2t^2-60t+500\), for \(5\le t\le25\), where \(t\) is the temperature of the sample in degree Celsius.
At what temperature does the number of bacteria reached the maximum?[3]
Find the maximum bacteria population.[2]
Given that \(y=\dfrac3{x^2+4}\) for \(x>0\), find \(\dfrac{\mathrm dy}{\mathrm dx}\) and explain why \(y\) is a decreasing function.[3]
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Answer:(a)(i) \(12^\circ\text C\). (ii) 1004 bacteria. (b) \(-\dfrac{6x}{(x^2+4)^2}<0\) for \(x>0\).