Question

Assume g is a real function such that g(x)∈O(xlogx). Indicate the least integer n, for which

f(x)=7x²g(x−1)+(x²+1)logx∈O(xⁿ)

Answer 1

The least integer n such that f(x)=7x²g(x-1)+(x²+1)logx is in O(xⁿ), given g(x) is in O(xlogx), is n = 3. The term 7x²g(x-1) is in O(x³), and (x²+1)logx is in O(x²logx), therefore f(x) is in O(x³).

Step-by-step explanation:

The student has asked to determine the least integer n such that f(x) = 7x²g(x-1) + (x²+1)logx is in O(xⁿ), given that g(x) is a function in O(xlogx). Since g(x) is in O(xlogx), g(x-1) can be considered to be in O((x-1)log(x-1)) because it does not change the asymptotic complexity.

By this property, the term 7x²g(x-1) will also be bounded by a function of the form x³ times a logarithmic factor, so it is in O(x³). The term (x²+1)logx is simply in O(x²logx). Since O(x³) encompasses O(x²logx), the whole function f(x) is in O(x³), making n = 3 the least integer for which f(x) is in O(xⁿ).