Local Maximum and Minimum Points
where x ∈ [−2, 3]. (a) What is f 0 (x)? [1] (b) What is f 00(x)? [1] (c) Find all the stationary points of f. Find the value of f(x) at each stationary point. [2] (d) For each stationary point of f, find whether it is a local minimum, a local maximum or neither. [2] (e) Find all the global minima and maxima of f in the interval [−2, 3]. [2] 2. Let M = 1 6 1 0 (a) Find the eigenvalues of M. [3] (b) Find one eigenvector of M with respect to each eigenvalue. [3]
(c) Diagonalize M, i.e., write M in the form M = NDN −1 , where D is a diagonal matrix. [2] 3. X is a continuous random variable with density f where f(x) = x + x 2 + x 5 for x ∈ [0, 1] and f(x) = 0 elsewhere. What is the probability of the event {X > 1/2}? [6] 4. Suppose you want to launch your startup in some innovative area. Currently, there exist two enterprises A and B competing for this market. You want to find out their investment strategies (which are given by real numbers a, b) prior to commiting into the competition.
A friend who works in A told you that their profit is given by p(a, b) = ab2 − a 2 + b 2 + 6b − 9a + 10, but unfortunately he does not know the values of a, b. He also mentioned that both enterprises are apparently not very good in keeping secrets, so the function f(x) = p(x, b) achieves the maximum value at point x = a, while x = b minimises g(x) = p(a, x). Use this information to find a and b [8] 5. A positive integer is called prime-like if it is not divisible by 2 and not divisible by 5.
(a) How many prime-like numbers with 4 digits (from 1000 to 9999)? [2] (b) How many prime-like numbers with 4 digits have at least one even digit? [3]
(c) How many prime-like numbers with 4 digits are not divisible by 3? [4] 6. Let Y be a random variable uniformly distributed on the set {−1, 0, 1, 2}. Let U1, U2 be the random variables defined by U1 = Y 2 − Y and U2 = Y 2−1 Y 3−2 . (a) Find the expected values of U1 and U2. [2] (b) Find variances of U1 and U2. [2] (c) Is U1 independent of U2? Justify your answer. [2] 7. A fair 6-sided die is rolled three times. (a) Find the probability that the sum of all three outcomes equals 8.
[4] (b) Find the probability that the first roll was 1 given that the sum of outcomes equals 8. [3] 8. Eight students Alice, Bob, Casey, Drew, Eva, Francis, Glen, Hunter entered Monash University this year. Alice, Bob and Casey are from Canberra and Drew, Eva, Francis are from Newcastle.
Glen and Hunter are international students from different countries. Alice, Casey and Hunter practice tennis, while Drew, Francis, Glen and Hunter practice soccer. Any two students from the same city or practicing the same sport know each other. (a) Draw a graph corresponding to the student acquaintances. Find its number of edges.
[2] (b) Write down the adjacency matrix for this graph. Does it contain a spanning tree? [3] (c) Is it possible to seat all eight students at a round table in such a way that any of them knows both neighbours. Justify your answer. [3]
Local Maximum and Minimum Points
-
- is the derivative of with respect to
power rule, subtraction and additional rule we obtain
For
is the derivative of with respect to
, addition and the subtraction rule we obtain
Since then
for
- The stationery points are at point
Hence the stationery points will be at the roots of
Using the quadratic equation
We obtain the roots of the function at the points
The value of at the stationery points will be
- Finding local minimum and local maximum
The stationery points are at
Now we use the sign test to determine if the points are maximum or minimum
Using -2 and 0 to do the sign test we can see that the sign changes from positive to negative. When the sign changes from positive to negative, then this indicates that we have a local maximum. Hence point (-1,7) is a local maximum point.
Now using the test sign, we test point (-2,6) using the values -2 and 0
The sign is changing from negative to positive hence the point (-2,6) is a local minimum.
- Graphing the function f gives
![]()
From this graph we can observe two points which can be classified as global maximum and minimum. That is point; (-1,7) and (-2,6)
-
- Eigenvalues of M
Using the characteristic polynomial
eigenvalues as
And
- Finding eigen vector
For every we find its own vectors
, so, we have a homogeneous system of linear equations we solve it by Gaussian Elimination
(1)
Find the variables from the equation of the system (1)
, we have a homogeneous system of linear equations, we solve it by Gaussian Elimination to obtain
(1)
- Diagonalize M
The diagonal matrix (the diagonal entries are the eigenvalues
The matrix with the eigenvectors ( as its columns
- Calculate probability
- The profit function is
The optimal points are at the derivative of the profit function equals to zero.
The derivative with respect to b gives
The derivative with respect to a
Now
Now
we obtain b
Since the value of b is minimum we take -3
This gives the value of a
- Not divisible by 2 and not divisible by 5.
- This means the last digit cannot be
Now we must pick 4 digits from a bucket of 10
This can be picked in the following way
The fist number can be picked in 9 ways
2nd in 10 ways
3rd in 10 ways
4th in 4 ways
Hence the total prime like numbers between 1000 and 9999 is
- Has at least one even digit
Here we select 4 numbers from a bucket of 10 in the following way
1st number 9 ways
2nd number 10 ways
3rd number 10ways
4th number 5 ways
Now let’s assume that the even digit is in the 1st number then the numbers will be
Leta now assume the even number is in the second digit
Let’s assume the even number is in the 3rd digit
The even digit cannot be in the fourth number as this will make the number prime like anymore
In total the numbers are
- Prime like numbers not divisible by 3
Here we must choose the four numbers such that the sum of the digit is not divisible by 3
1st digit 9 ways
2nd digit 10 ways
3rd digit 10 ways
4th digit 3 ways
The numbers are
- Y uniformly distribute
Since Y is uniformly distributed all the values occur with equal probability the expected value of
expected value of
For
Since all the values occur with uniform probability the expected value will be
- Variance of
The variance is 0
Variance of
The variance is 0
- Testing independence of and
No, the equations are dependent as they share similar variance
- A fair 6-sided dice is rolled 3 times
- Probability that sum is 8
Possible outcomes are
Each roll must have a value between 1 and 6
The only way we can have a sum of 8 is
The roll is 3 times so there are
Thus, the probability is
- Probability of 1 given sum of 8
- Drawing graph
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- The adjacency matrix is given by the graph does not have a spanning tree.
- No, there is no connection that can allow all eight students to seat in a way that any of them knows both neighbours.
References
McQuarrie, D., 2003. Mathematical Methods for Scientists and Engineers, s.l.: University Science Books.
Salas, S. L., Hille, E. & Etgen, G. J., 2007. Calculus: One and Several Variables. 10th ed. s.l.:Wiley.
