5.10 Second derivative and concavityIB Maths: Applications and Interpretation HL: Flashcards

What these 12 flashcards ask

Two notations for the second derivative?
What does f''(x)0 mean for the graph?
What does f''(x)<0 mean for the graph?
State the second derivative test for a maximum.
State the second derivative test for a minimum.
What if f''(x)=0 at a stationary point?
What is a point of inflexion?
Is f''(x)=0 enough for an inflexion?
In context, what does an inflexion on a growth model show?
What is acceleration in terms of displacement?
Find f''(x) for f(x)=x^3-6x^2.
Can a graph be increasing and concave-down?

Exam questions on 5.10 Second derivative and concavity

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
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
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

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).