Question

41. Find the least integer n such that f(x) is O(x^n) for each of these functions(a) f(x) = 2x2 + x^3 logx​

Answer 1

Answer: - In the given question, the result will be - f(x) = O(x⁴) with k = 2 and C = 2.

Detailed solution: -

The function given is: -

f(x) = 2x² + x³ log x

• We need to find the value of n for the given function to satisfy the O(xⁿ) notation.
• The Big notation O represents the maximum time of operating the algorithm.

Hence,

• The property here, log x ≤ x holds in the given equation when x > 0.

Therefore,

f(x) = 2x² + x³ log x ≤ 2x² + x⁴

• The maximum power of the x in the given expression of the f(x) is equal to the smallest n for which f(x) is equal to O(xⁿ).

Therefore,

n = 4

When x > 2, then we have the property as x² > x > 2.

Ιf(x)Ι = Ι2x² + x³ log xΙ ≤ Ι2x² + x⁴Ι ≤ Ι2x²Ι + Ιx⁴Ι

= 2x² + x⁴ ≤ x⁴ + x⁴

= 2x⁴

=2Ιx⁴Ι

Therefore,

C should be at least 2.

Let us know choose the value of C = 2.

Hence,

f(x) = O(x⁴) with k = 2 and C = 2.

Therefore,

• The answer is f(x) = O(x⁴) with k = 2 and C = 2.

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