CAIE IGCSE Additional Math 14.9 Use Derivative Tests for Maxima and Minima Questions

Practise applying first- and second-derivative tests to classify stationary points as maxima or minima in Cambridge Additional Mathematics problems.

Exam points

Use first- and second-derivative tests to classify stationary points and determine whether they are maxima or minima.

[Maximum number: 5]

A cylinder has radius $r \mathrm{~cm}$ and height $h \mathrm{~cm}$ .
The total surface area, including the two ends, is $A \mathrm{~cm}^{2}$ .
The volume of the cylinder is $330 \mathrm{~cm}^{3}$ .

Given that r can vary, find the value of r that gives a stationary value for A and show that this value is a minimum.

Correct derivative: $4\pi r-660r^{-2}$ , isw
B1 for one correct term.
$4\pi r-\frac{660}{r^2}=0$ and solves for r
M1 FT their derivative, providing at least one term is correct.
$r=\sqrt[3]{\frac{660}{4\pi}}$ oe, isw, or 3.74 or 3.74494... rot to 3 or more dp
Second derivative $\frac{d^2S}{dr^2}=4\pi+1320r^{-3}$ , and when r=3.74, $\frac{d^2S}{dr^2}>0$ oe, hence minimum.
A1 dep on previous mark.

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CAIE IGCSE Additional Math 14.9 Use Derivative Tests for Maxima and Minima Questions question 1