Module MA1132 (Frolov), Advanced Calculus
Homework Sheet 6
Each set of homework questions is worth 100 marks
Due: at the beginning of the tutorial session Thursday/Friday, 10/11 March 2016
Name:
1. Consider the function
f (x, y) = x2 − xy 2 − 3x + y 4 + 5
Locate all relative maxima, relative minima, and saddle points, if any. Use Mathematica
to plot its graph.
2. Consider the intersection of the surfaces
p
x2
y2
2
2
2
+ 2 = 1, a > b > c.
z = a − x − y , and
b2
c
p
p
(a) What is the surface z = a2 − x2 − y 2 ? Sketch the surface z = a2 − x2 − y 2 and
its projection onto the xy plane for a = 3.
2
2
(b) What is the surface xb2 + yc2 = 1 ? Sketch the surface
onto the xy plane for b = 2, c = 1.
x2
b2
+
y2
c2
= 1 and its projection
(c) Use Lagrange multipliers to find the coordinates of the points on the intersection
which have the maximum z-coordinate and the minimum z-coordinate.
3. Let an open top box (a box without a lid) have volume 4m3 . Find the dimensions of the
box so that the area A of the box is minimised. What is the minimum area A?
4. Show that a triangle with fixed area has minimum perimeter if it is equilateral.
5. Consider the intersection of the surfaces
z = λ x + µ y + h , λ > 0 , µ > 0 , h > 0 , and
x2
y2
+ 2 = 1.
a2
b
(a) What is the surface z = λ x + µ y + h ? What is the surface
x2
a2
+
y2
b2
=1?
(b) Sketch the surfaces for a = b = λ = µ = h = 1.
(c) Use the Lagrange multiplier method to find the coordinates of the points on the
intersection which have the maximum z-coordinate and the minimum z-coordinate.
6. What is the volume of the largest n-dimensional box with edges parallel to the coordinate
axes that fits inside the n-dimensional ellipsoid
x21 x22
x2n
+
+ ··· + 2 = 1.
a21 a22
an
(1)
Bonus questions (each bonus question is worth extra 25 marks)
1. Given three points P1 (a1 , b1 ), P2 (a2 , b2 ), P3 (a3 , b3 ) on a plane find all points which have
the smallest sum of distances from the three points.
1