To analyze the zeroes of the function [tex]\(f(x) = -x^5 + 9x^4 - 18x^3\)[/tex], we aim to determine the points where the function equals zero. Let's carefully consider the roots and their multiplicities based on the detailed solution provided:
### Step 1: Finding the roots
The roots of the function are the values of [tex]\(x\)[/tex] at which [tex]\(f(x) = 0\)[/tex].
- The given solution identifies the roots of the function as [tex]\(0\)[/tex], [tex]\(3\)[/tex], and [tex]\(6\)[/tex].
### Step 2: Determining the multiplicities
The multiplicity of a root indicates how many times that root appears as a solution of the equation.
- The multiplicities of the roots found are:
- Root [tex]\(0\)[/tex] has a multiplicity of [tex]\(3\)[/tex].
- Root [tex]\(3\)[/tex] has a multiplicity of [tex]\(1\)[/tex].
- Root [tex]\(6\)[/tex] has a multiplicity of [tex]\(1\)[/tex].
### Conclusion:
Based on the given information:
- The root [tex]\(0\)[/tex] has multiplicity [tex]\(3\)[/tex].
- The root [tex]\(3\)[/tex] has multiplicity [tex]\(1\)[/tex].
- The root [tex]\(6\)[/tex] has multiplicity [tex]\(1\)[/tex].
Thus, the correct description of the zeroes of the graph of [tex]\(f(x) = -x^5 + 9x^4 - 18x^3\)[/tex] is:
[tex]\[ 0 \text{ with multiplicity } 3, 3 \text{ with multiplicity } 1, \text{ and } 6 \text{ with multiplicity } 1. \][/tex]
This matches with the option:
[tex]\[ \boxed{0 \text{ with multiplicity 3, 3 with multiplicity 1, and 6 with multiplicity 1}} \][/tex]
