To analyze the behavior of the graph of the function \( f(x) = x(-x + 6)(-x - 1)^2 \) at its zeros, we first need to find the zeros and the nature of each zero:
- Finding the Zeros:
- The term \( x = 0 \) gives a zero at \( x = 0 \).
- The term \( -x + 6 = 0 \) gives \( x = 6 \).
- The term \( (-x - 1)^2 = 0 \) gives a zero at \( x = -1 \) with a multiplicity of 2 (since it is squared).
So, the zeros of the function \( f(x) \) are at \( x = 0 \), \( x = 6 \), and \( x = -1 \).
- Nature of the Zeros:
- The zero at \( x = 0 \) has multiplicity 1, which means the graph will cross the x-axis at this point.
- The zero at \( x = 6 \) also has multiplicity 1, meaning the graph will cross the x-axis at this point as well.
- The zero at \( x = -1 \) has multiplicity 2, which means the graph will touch the x-axis and turn around at this point.
Now, we can summarize the behavior:
- The graph crosses the x-axis at \( x = 0 \).
- The graph crosses the x-axis at \( x = 6 \).
- The graph touches the x-axis at \( x = -1 \) and turns around.
Given this information, the correct response is:
The graph crosses the x-axis at 0 and 6, and touches the x-axis and turns around at -1.