APM4805
ASSIGNMENT 1 2024
DUE DATE: 31 May 2024
, ASSIGNMENTS
Instructions for the Assignments
Take care to explain all your arguments.
Only PDF …les will be accepted.
ASSIGNMENT 01
Due date: Friday, 31 May 2024
Note: Answer the following questions 1 to 4 related to the Study Guide APM4805/102/0/2024,
Exercises section 1.5.
Question 1. Investigate the maxima and minima of the following functions over the real line:
(a) f (x) = 2x 2 + 3
(b) f (x) = jx 2j + jx 1j
(c) f (x) = e 1 x
2
(d) f (x) = x
x
[20 marks]
Question 2. Investigate the minima and maxima of f (x; y) = 3x + 2y 1 on the following sets:
(a) x 2 + y 2 1
(b) x 0, y 0
[10 marks]
Question 3. Find the following:
(a) inf(e x + e x ) on R
(b) sup e jxj on R
(c) The level sets S0 and S5 for S = R, f (x) = e jxj .
(d) The level sets S1 and S2 for S = f(x; y) : jxj + jyj 1g, f (x) = e jxj+jyj .
[20 marks]
Question 4. Find the level curves ff (x; y) = cg of each of the following functions f through the two points (0; 0) and (1; 2),
and determine the sets ff (x; y) < cg and ff (x; y) > cg:
(a) f (x; y) = x 2 + y 2
(b) f (x; y) = xy
[10 marks]
Note: Answer the following questions 5 to 8 related to the Study Guide APM4805/102/0/2024,
Exercises section 3.7.
Question 5. Find the critical points and critical values of the following functions, and determine which critical points
determine local extrema:
(a) f (x; y) = x 2 + y 2 + 4,
(b) f (x; y) = x 2 y2 + xy
[10 marks]
1
, Question 6. Consider the function f : R 2 ! R determined by
1 2 2
f (x) = x T x+xT + 2:
2 4 3
(a) Find the gradient and Hessian of f at the point (1; 1).
(b) Find the directional derivative of f at (1; 1) in the direction of the maximal rate of increase.
(c) Find a point that satis…es the …rst order necessary condition. Does the point also satisfy the second order necessary
condition for a minimum?
[15 marks]
2 2
Question 7. Find the critical points of the function f (x; y) = x 4 + y 2:
Show that f has a global minimum at each of the points ( x; y) = (2; 0) and (x; y) = ( 2; 0). Show that the point (0 ; 0) is a
saddle point. Sketch the level curves f (x; y) = constant for selected values of the constant.
[15 marks]
ax 2 + 2bxy + cy 2
Question 8. Find the critical points and critical values of the function f (x; y) = .
x2 + y 2
a b
Show that the critical values are solutions of the equation b c = 0:
[10 marks]
[Total: 100 marks]
–End of assignment –
2
,
ASSIGNMENT 1 2024
DUE DATE: 31 May 2024
, ASSIGNMENTS
Instructions for the Assignments
Take care to explain all your arguments.
Only PDF …les will be accepted.
ASSIGNMENT 01
Due date: Friday, 31 May 2024
Note: Answer the following questions 1 to 4 related to the Study Guide APM4805/102/0/2024,
Exercises section 1.5.
Question 1. Investigate the maxima and minima of the following functions over the real line:
(a) f (x) = 2x 2 + 3
(b) f (x) = jx 2j + jx 1j
(c) f (x) = e 1 x
2
(d) f (x) = x
x
[20 marks]
Question 2. Investigate the minima and maxima of f (x; y) = 3x + 2y 1 on the following sets:
(a) x 2 + y 2 1
(b) x 0, y 0
[10 marks]
Question 3. Find the following:
(a) inf(e x + e x ) on R
(b) sup e jxj on R
(c) The level sets S0 and S5 for S = R, f (x) = e jxj .
(d) The level sets S1 and S2 for S = f(x; y) : jxj + jyj 1g, f (x) = e jxj+jyj .
[20 marks]
Question 4. Find the level curves ff (x; y) = cg of each of the following functions f through the two points (0; 0) and (1; 2),
and determine the sets ff (x; y) < cg and ff (x; y) > cg:
(a) f (x; y) = x 2 + y 2
(b) f (x; y) = xy
[10 marks]
Note: Answer the following questions 5 to 8 related to the Study Guide APM4805/102/0/2024,
Exercises section 3.7.
Question 5. Find the critical points and critical values of the following functions, and determine which critical points
determine local extrema:
(a) f (x; y) = x 2 + y 2 + 4,
(b) f (x; y) = x 2 y2 + xy
[10 marks]
1
, Question 6. Consider the function f : R 2 ! R determined by
1 2 2
f (x) = x T x+xT + 2:
2 4 3
(a) Find the gradient and Hessian of f at the point (1; 1).
(b) Find the directional derivative of f at (1; 1) in the direction of the maximal rate of increase.
(c) Find a point that satis…es the …rst order necessary condition. Does the point also satisfy the second order necessary
condition for a minimum?
[15 marks]
2 2
Question 7. Find the critical points of the function f (x; y) = x 4 + y 2:
Show that f has a global minimum at each of the points ( x; y) = (2; 0) and (x; y) = ( 2; 0). Show that the point (0 ; 0) is a
saddle point. Sketch the level curves f (x; y) = constant for selected values of the constant.
[15 marks]
ax 2 + 2bxy + cy 2
Question 8. Find the critical points and critical values of the function f (x; y) = .
x2 + y 2
a b
Show that the critical values are solutions of the equation b c = 0:
[10 marks]
[Total: 100 marks]
–End of assignment –
2
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