Module MA1132 (Frolov), Advanced Calculus
Tutorial Sheet 6
To be solved during the tutorial session Thursday/Friday, 25/26 February 2016
1. Consider the function
f (x, y) = −x4 + 2x2 y − 2y 2 + 2y + 5
Locate all relative maxima, relative minima, and saddle points, if any.
2. What is the volume of the largest box with edges parallel to the coordinate axes that fits
inside the ellipsoid
x2 y 2 z 2
+ 2 + 2 = 1.
(1)
a2
b
c
3. Let (xi , yi ) , xi 6= xj , i, j = 1, . . . , n be n points on the xy-plane.
(a) Find the least squares line of best fit, that is the line y = ax + b which minimises
the vertical distance f (a, b),
n
X
f (a, b) =
(axi + b − yi )2 ,
(2)
i=1
between the points (xi , yi ) and the line.
P
(b) The surface z = f (a, b) = ni=1 (axi + b − yi )2 is an elliptic paraboloid, that is f (a, b)
can be written in the form
f (a, b) = A0 a02 + C 0 b02 + D0 a0 + E 0 b0 + F 0 ,
A0 C 0 > 0 ,
(3)
where
a0 = a cos φ + b sin φ ,
b0 = b cos φ − a sin φ .
(4)
Find A0 , C 0 , and explain how this shows that f (a, b) has an absolute minimum at
the point found in part (a) of this question.
1