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(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

3x2 + 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

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