Question

the graph of $f(x) = x^3 - 6x^2 + 9x$ is shown. based on the graph, what are the solutions of the equation $x^3 - 6x^2 + 9x = 0$? $\bigcirc$ $x = 3$ $\bigcirc$ $x = -3, 0$ $\bigcirc$ $x = 0, 3$ $\bigcirc$ $x = -3, 0, 3$

Explanation:

Step1: Recall root definition

The solutions of \(x^3 - 6x^2 + 9x = 0\) are the x - intercepts of \(f(x)=x^3 - 6x^2 + 9x\) (where \(y = 0\)).

Step2: Analyze the graph

From the graph, the curve intersects the x - axis at \(x = 0\) and touches (and turns around) at \(x = 3\) (which is also a root, a repeated root in this case). So the solutions are \(x = 0\) and \(x = 3\).

Answer:

C. \(x = 0, 3\)