Discrete Math f(x) = 2x + 3 is big-O of X2 . This means that there exist constan

Question

Discrete Math

f(x) = 2x + 3 is big-O of X2 . This means that there exist constants C and k such that If(x)| C lg (x) 1 for all x k. We learned in the lecture that in this case, there are always infinitely many suitable pairs (C,k) a. Find the lowest k that works with C 1 and justify your answer fully. Note that you have to show two aspects: that your k works, and that no lower k value can work. b. Find the lowest C that works with k 2 and justify your answer fully. You may use the calculus fact that if two continuous functions p and q satisfy p(x) k, they must also satisfy that inequality at x = k. Note that you have to show two parts: that your Cworks, and that no lower C value can work.

Explanation / Answer

We have f(x) = 2x+3 and g(x) = x2

We have been given that |f(x)|<=C|g(x)| for all x>k

Part (a)

We need to determine the minimal value of k for which C=1

Therefore, we set:

f(x)|<=C|g(x)|

|2x+3|<=x2

This implies two scenarios:

2x+3<=x2 or -(2x+3)<=x2

We get two inequalities as:

x2 - 2x -3 >=0 or x2 + 2x+ 3 >=0

x2 - 2x -3 >=0 gives us x>=3

x2 + 2x+ 3 >=0 gives us x can have any real number.

Therefore, the lowest value of k that we can have is 3.

Part (b)

In this part, we have to find the lowest value of C that works for k=2

We get:

f(x)|<=C|g(x)|

|2x+3|<=Cx2

We know solution of this inequality is x>=2

Therefore, lowest value of C will occur at x=2

|2*2+3|<=C22

C>= 7/4

Therefore, lowest value for C is 7/4

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