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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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