Question

Show that 10000x^2 is O(x^3) but x^3 is not O(10000x^2)

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1. To show \(10000x^2\) is \(O(x^3)\):

f(x) = 10000x^2

g(x) = x^3

Find \(C\) and \(k\) such that for all \(x > k\),

10000x^2 \leq Cx^3

\frac{10000}{x} \leq C

As \(x \to \infty\), \(\frac{10000}{x} \to 0\). Choose \(C = 10000\) and \(k = 1\). Thus,

10000x^2 \text { is } O(x^3).

2. To show \(x^3\) is not \(O(10000x^2)\):

f(x) = x^3

g(x) = 10000x^2

If \(x^3\) were \(O(10000x^2)\), for some \(C\) and \(k\),

x^3 \leq C \cdot 10000x^2

\frac{x}{10000} \leq C

As \(x \to \infty\), \(x\) cannot be bounded by a constant \(10000C\). Thus,

x^3 \text { is not } O(10000x^2).

Final answer:

10000x^2 \text { is } O(x^3) \text { but } x^3 \text { is not } O(10000x^2) .

Find \(C\) and \(k\) such that for all \(x > k\),

As \(x \to \infty\), \(\frac{10000}{x} \to 0\). Choose \(C = 10000\) and \(k = 1\). Thus,

2. To show \(x^3\) is not \(O(10000x^2)\):

If \(x^3\) were \(O(10000x^2)\), for some \(C\) and \(k\),

As \(x \to \infty\), \(x\) cannot be bounded by a constant \(10000C\). Thus,

Final answer:

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