Final answer:
The smallest integer n for the function f(x) to be Big-O of x^n is 5, because x^5 is the dominant term. Therefore, the correct answer is a) n = 5.
Step-by-step explanation:
The student is asking to find the smallest integer n for which the function f(x) = x^5 - x^3 log(x) is Big-O of x^n. To determine the Big-O notation, we typically look for the highest power of x that has the greatest growth rate as x approaches infinity. The term x^5 grows faster than x^3 log(x) as x approaches infinity. Thus, the dominant term is x^5, making f(x) Big-O of x^5. Therefore, the correct answer is n = 5, which corresponds to option (a).
We can ignore the second term, x^3 log(x), in comparison to the x^5 term when considering Big-O notation because the logarithmic function grows much more slowly than any polynomial term and does not affect the leading term's degree for large values of x. To find the smallest integer n for which f(x) = x^5 - x^3 log(x) is Big-O of x^n, we need to determine the highest power of x in the equation. The term x^5 has the highest power, so n = 5. Therefore, the correct answer is a) n = 5.