To find the multiplicity of \((3x + 1)\) in the function \(k(x) = -0.5x^5(3x + 1)^2(1 - 4x)\), we look at the factor \((3x + 1)\) in the expression.
The factor appears as \((3x + 1)^2\), which indicates that the multiplicity of the root associated with \((3x + 1)\) is 2.
When determining the behavior of the graph at the associated x-intercept, we consider the following rules:
- If the multiplicity is odd, the graph crosses the x-axis at that point.
- If the multiplicity is even, the graph touches the x-axis and turns around.
Since the multiplicity of \((3x + 1)\) is 2 (which is even), the graph of \(k(x)\) will touch the x-axis at the x-intercept associated with this factor and turn around.
Thus, the correct answer is:
The multiplicity of \((3x + 1)\) is 2. At the associated x-intercept, the graph of \(k(x)\) touches the x-axis and turns around.