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 $y=0$: $-\frac{x^3}{3}+x^2=0 \Rightarrow x^2(-\frac{x}{3}+1)=0 \Rightarrow x=0,3$.
Intercepts: $(0,0),(3,0)$.

Step2: Find critical points

Compute $y'=-x^2+2x$. Set $y'=0$: $-x(x-2)=0 \Rightarrow x=0,2$.

Step3: Determine increasing/decreasing

Test intervals:

• $x<0$: $y'<0$ (decreasing)
• $00$ (increasing)
• $x>2$: $y'<0$ (decreasing)

Intervals: Decreasing $(-\infty,0)\cup(2,\infty)$; Increasing $(0,2)$.

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

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

Step6: Determine concavity

Test intervals:

• $x<1$: $y''>0$ (concave up)
• $x>1$: $y''<0$ (concave down)

Intervals: Concave up $(-\infty,1)$; Concave down $(1,\infty)$.

Step7: Sketch the graph

Plot intercepts $(0,0),(3,0)$; extrema $(0,0)$ (min), $(2,\frac{4}{3})$ (max); inflection point $(1,-\frac{1}{3}+1=\frac{2}{3})$. Connect points with increasing/decreasing and concavity.

Answer:

Intercepts: $(0,0)$, $(3,0)$
Critical points: $x=0$, $x=2$
Increasing interval: $(0,2)$
Decreasing intervals: $(-\infty,0)$, $(2,\infty)$
Relative minimum: $(0,0)$
Relative maximum: $(2,\frac{4}{3})$
Inflection point: $x=1$
Concave up interval: $(-\infty,1)$
Concave down interval: $(1,\infty)$
Graph: Plotted using above points and intervals.