for each problem, find the x and y - intercepts, x - coordinates of the critical points, open intervals…

Explanation:

Step1: Find x - intercepts

Set $y = 0$, so $-\frac{x^{3}}{3}+x^{2}=0$. Factor out $x^{2}$: $x^{2}( -\frac{x}{3}+1)=0$. Then $x = 0$ or $x = 3$.

Step2: Find y - intercepts

Set $x = 0$, then $y=-\frac{0^{3}}{3}+0^{2}=0$.

Step3: Find the first - derivative

Differentiate $y =-\frac{x^{3}}{3}+x^{2}$ with respect to $x$. Using the power rule $y^\prime=-x^{2}+2x$.

Step4: Find critical points

Set $y^\prime = 0$, so $-x^{2}+2x = 0$. Factor out $-x$: $-x(x - 2)=0$. The critical points are $x = 0$ and $x = 2$.

Step5: Determine intervals of increase and decrease

Test intervals: For $x\lt0$, let $x=-1$, then $y^\prime=-(-1)^{2}+2(-1)=-3\lt0$, so the function is decreasing on $(-\infty,0)$. For $0\lt x\lt2$, let $x = 1$, then $y^\prime=-1^{2}+2\times1 = 1\gt0$, so the function is increasing on $(0,2)$. For $x\gt2$, let $x = 3$, then $y^\prime=-3^{2}+2\times3=-3\lt0$, so the function is decreasing on $(2,\infty)$.

Step6: Find the second - derivative

Differentiate $y^\prime=-x^{2}+2x$ with respect to $x$. $y^{\prime\prime}=-2x + 2$.

Step7: Find inflection points

Set $y^{\prime\prime}=0$, so $-2x + 2=0$, which gives $x = 1$.

Step8: Determine concavity

For $x\lt1$, let $x = 0$, then $y^{\prime\prime}=-2\times0 + 2=2\gt0$, so the function is concave up on $(-\infty,1)$. For $x\gt1$, let $x = 2$, then $y^{\prime\prime}=-2\times2+2=-2\lt0$, so the function is concave down on $(1,\infty)$.

Step9: Find relative extrema

Since the function changes from decreasing to increasing at $x = 0$, $y(0)=0$ is a relative minimum. Since the function changes from increasing to decreasing at $x = 2$, $y(2)=-\frac{2^{3}}{3}+2^{2}=-\frac{8}{3}+4=\frac{4}{3}$ is a relative maximum.

To sketch the graph: Mark the x - intercepts at $x = 0$ and $x = 3$, the y - intercept at $(0,0)$, the critical points at $(0,0)$ and $(2,\frac{4}{3})$, and the inflection point at $(1,\frac{2}{3})$. Then draw the curve according to the intervals of increase/decrease and concavity.

Answer:

x - intercepts: $x = 0,x = 3$; y - intercept: $(0,0)$; critical points: $x = 0,x = 2$; increasing on $(0,2)$, decreasing on $(-\infty,0)\cup(2,\infty)$; inflection point: $x = 1$; concave up on $(-\infty,1)$, concave down on $(1,\infty)$; relative minimum at $(0,0)$, relative maximum at $(2,\frac{4}{3})$