Question

Consider the function f(x)=6x² (x-2)³-2-3 over the interval [1,8]. Does the extreme value theorem guarantee the existence of an absolute maximum and minimum for f(x) on this interval?

Answer 1

The Extreme Value Theorem guarantees the existence of both an absolute maximum and minimum for the function f(x) on the closed interval [1,8], since this polynomial function is continuous on that interval.

Step-by-step explanation:

The Extreme Value Theorem states that if a function is continuous on a closed interval, then it must have both an absolute maximum and an absolute minimum on that interval. Given the function f(x)=6x²(x-2)³-2-3, we need to ascertain the continuity of the function on the interval [1,8]. A polynomial function is inherently continuous everywhere; thus, because our interval is closed and the function f(x) is a polynomial, we can conclude that f(x) indeed has both an absolute maximum and an absolute minimum on the interval [1,8], according to the Extreme Value Theorem.