5.10 Second derivative and concavityIB Maths: Applications and Interpretation HL: Mind map

Exam questions on 5.10 Second derivative and concavity

1. The profit, $P$ thousand euros, of a company in year $t$ is modelled by $P(t)=t^3-9t^2+15t+20$ for $0\le t\le8$.
   Find the time at which the profit is falling most rapidly, and the rate of change of the profit at that time.2 marks
2. A factory's average cost per unit, $C$ euros, when it makes $x$ hundred units is modelled by $C(x)=x+\dfrac{16}{x}$ for $x>0$.
   Find the number of hundreds of units that gives the least average cost, using the second derivative test to justify your answer.2 marks
3. The concentration of a drug in a patient's blood, $c$ mg l$^{-1}$, is modelled by $c(t)=4te^{-t/2}$ for $t\ge0$, where $t$ is the time in hours after the drug is given.
   Find $c'(t)$ and hence show that $c''(t)=(t-4)e^{-t/2}$.3 marks