Question

for each problem, find the: x and y intercepts, x - coordinates of the critical points, open intervals where the function is increasing and decreasing, x - coordinates of the inflection points, open intervals where the function is concave up and concave down, and relative minima and maxima. using this information, sketch the graph of the function.

1. ( y = -\frac{x^{3}}{3}+x^{2} )

Explanation:

Step1: Find intercepts

Set $x=0$: $y=0$. Set $y=0$: $-\frac{x^3}{3}+x^2=0\Rightarrow x^2(3-x)=0\Rightarrow x=0,3$.

Step2: Find critical points

First derivative: $y'=-x^2+2x=-x(x-2)$. Set $y'=0$: $x=0,2$.

Step3: Determine increasing/decreasing

Test intervals: $(-\infty,0)$: $y'<0$ (decreasing); $(0,2)$: $y'>0$ (increasing); $(2,\infty)$: $y'<0$ (decreasing).

Step4: Find relative extrema

At $x=0$: decreasing→increasing (min, $y=0$); at $x=2$: increasing→decreasing (max, $y=-\frac{8}{3}+4=\frac{4}{3}$).

Step5: Find inflection points

Second derivative: $y''=-2x+2$. Set $y''=0$: $x=1$.

Step6: Determine concavity

Test intervals: $(-\infty,1)$: $y''>0$ (concave up); $(1,\infty)$: $y''<0$ (concave down).

Step7: Sketch the graph

Plot intercepts $(0,0),(3,0)$, extrema $(0,0),(2,\frac{4}{3})$, inflection point $(1,-\frac{1}{3}+1=\frac{2}{3})$, then connect with curves matching increasing/decreasing and concavity.

Answer:

Intercepts: $(0,0),(3,0)$; Critical points: $x=0,2$; Increasing: $(0,2)$; Decreasing: $(-\infty,0)\cup(2,\infty)$; Inflection point: $x=1$; Concave up: $(-\infty,1)$; Concave down: $(1,\infty)$; Relative min: $(0,0)$; Relative max: $(2,\frac{4}{3})$; Graph sketched using these points and behaviors.