(solution) Mathematics IB Tutorial 8 (week 9) Semester 2, 2016 1. Consider

(solution) Mathematics IB Tutorial 8 (week 9) Semester 2, 2016 1. Consider

the questions are via attachment. can yo please solve all the questions.

Mathematics IB Tutorial 8 (week 9)
Semester 2, 2016
1. Consider the quadratic equation
3×2 + 4y 2 + 5z 2 + 4xy ? 4yz = 1
(a) Write the left hand side as xt Ax for some 3 × 3 symmetric matrix
A.
(b) The matrix A has characteristic polynomial (? ? 1)(? ? 4)(? ? 7).
Calculate the eigenvectors of A and hence write down a matrix
P that orthogonally diagonalises A.
(c) Use this matrix P to make a change of variables and put the
equation into standard form. What quadric surface does it represent?
(
x2 + 3x + 2 if x < 0
2. Let f (x) =
.
3x ? x3
if x ? 0
(a) Find all critical points of f .
(b) Find the intervals on which f is increasing and those on which f
is decreasing.
(c) Find the intervals on which the graph of f is concave up and
where it is concave down.
You must justify your answers by referring to theorems from lectures. 1