MA208 Optimisation Theory
Exercises 7 (First-order conditions)
Exercise 7.1.
Consider the following functions f,g,h:R2→Rand decide whether they attain a
global minimum or global maximum. If they do, determine where it is attained.
(a) f(x,y) = 4x2+xy3+y2,
(b) g(x,y) = x/(1+x2+y2),
(c) h(x,y) = ycos x.
Exercise 7.2.
Let f:R≥→Rbe a C1function so that f(0) = 0, f(1)>0, and limx→∞f(x) = 0.
Suppose there is only a single x∗∈R≥at which f0(x∗) = 0. Show that
(a) x∗is global maximizer of fon R≥, and that
(b) f(x)≥0 for all x≥0.
Exercise 7.3.
A manufacturer of aluminium beer cans has to produce cylindrical cans where top
and bottom are discs with radius rand the sidewall of the cylinder has height h. The
sidewall has thickness 1 and the top and bottom have thickness Afor some parameter
A>0 (both 1 and Aare very small compared to rand h). The prescribed volume of the
cylinder is V>0 (we neglect the volume of the aluminium needed for the can itself).
With the help of the Theorem of Lagrange, find rand hso that the can is made with the
least amount of material. The following picture shows a layout of the material needed.
h
r
top and bottom:
thickness
cylinder sidewall:
thickness 1
A
How are hand rrelated when A=1? What are rand hif V=324 cm3and A=6/π≈
1.91?
c
LSE 2016 /MA208 Page 1 of 1
