Question

analyze the polynomial function ( f(x)=x^{2}(x - 14) ) using parts (a) through (h) below.
(e) approximate the turning points of the graph.
the turning point(s) of the graph is/are (0.00,0.00),(9.33, - 406.52).
(type an ordered pair. round each coordinate to the nearest hundredth as needed. use a comma to
separate answers as needed.)
(f) use the information in parts (a) through (e) to draw a complete graph of the function by hand.
choose the correct graph below.

Explanation:

Step1: Analyze the function \( f(x)=x^{2}(x - 14)=x^{3}-14x^{2} \)

The degree of the polynomial \( y = f(x)\) is \(n = 3\) (odd), and the leading coefficient \(a=1>0\). As \(x\to-\infty\), \(y\to-\infty\); as \(x\to+\infty\), \(y\to+\infty\).

Step2: Find the \(x -\)intercepts

Set \(y = 0\), then \(x^{2}(x - 14)=0\). The solutions are \(x = 0\) (with multiplicity \(2\), the graph touches the \(x -\)axis at \(x = 0\)) and \(x=14\) (with multiplicity \(1\), the graph crosses the \(x -\)axis at \(x = 14\)).

Step3: Analyze the general shape of the graph

Since the function is a cubic polynomial \(y=x^{3}-14x^{2}\), and we know the end - behavior and \(x -\)intercepts. When \(x = 0\), \(y=0\); when \(x=14\), \(y = 0\).
We can also find the derivative \(y^\prime=f^\prime(x)=3x^{2}-28x=x(3x - 28)\). The critical points are \(x = 0\) and \(x=\frac{28}{3}\approx9.33\).
When \(x = 0\), \(y = 0\); when \(x=\frac{28}{3}\), \(y=(\frac{28}{3})^{2}(\frac{28}{3}-14)=(\frac{784}{9})(-\frac{14}{3})=-\frac{10976}{27}\approx - 406.52\)

Answer:

B.