Answer
(a)
$x$-intercepts: $0$ and $4.5$
$y$-intercept: $0$
(b)
relative minimum is at $(3, -3)$
relative maximum is at $$(0, 0)$
Work Step by Step
(a)
The $x$-intercepts can be found by looking at the points where the graph touches/crosses the $x$-axis.
However, one of the $x$-intercepts is between two integers.
To find the exact value of the $x$-intercepts, fund it algebraically by setting $P(x)=0$ then solving for $x$:
\begin{align*}
\frac{2}{9}x^3-x^2&=0\\
x^2\left(\frac{2}{9}x-1\right)&=0\\
\end{align*}
Use the Zero-Product Property by equating each factor to zero, then solve each equation to obtain:
\begin{align*}
x^2&=0 &\text{or}& &\frac{2}{9}x-1=0\\\\
x&=0 &\text{or}& &\frac{2}{9}x=1\\\\
x&=0 &\text{or}& &\frac{9}{2}\cdot \frac{2}{9}x=1\cdot \frac{9}{2}\\\\
x&=0 &\text{or}& &x=\frac{9}{2}
\end{align*}
Thus, the $x$-intercepts are: $0$ and $4.5$
The graph crosses the $y$-axis at $(0,0)$.
Thus, the $y$-intercept is $0$.
(b)
The graph has a relative maximum at $(0,0)$ and a relative minimum at $(3, -3)$.
