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 $x=0$: $y=0$ → y-intercept $(0,0)$.
Set $y=0$: $-\frac{x^3}{3}+x^2=0 \Rightarrow x^2(-\frac{x}{3}+1)=0 \Rightarrow x=0$ or $x=3$ → x-intercepts $(0,0),(3,0)$.

Step2: Find critical points

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

Step3: Increasing/decreasing intervals

Test intervals:

• $(-\infty,0)$: $y'<0$ → decreasing.
• $(0,2)$: $y'>0$ → increasing.
• $(2,\infty)$: $y'<0$ → decreasing.

Step4: Relative extrema

At $x=0$: decreasing→increasing → relative minimum.
At $x=2$: increasing→decreasing → relative maximum.

Step5: Find inflection points

Second derivative: $y''=-2x+2=-2(x-1)$.
Set $y''=0$: $x=1$ → inflection x-coordinate $1$.

Step6: Concavity intervals

Test intervals:

• $(-\infty,1)$: $y''>0$ → concave up.
• $(1,\infty)$: $y''<0$ → concave down.

Answer:

• x-intercepts: $0,3$; y-intercept: $0$
• Critical x-coordinates: $0,2$
• Increasing: $(0,2)$; Decreasing: $(-\infty,0)\cup(2,\infty)$
• Inflection x-coordinate: $1$
• Concave up: $(-\infty,1)$; Concave down: $(1,\infty)$
• Relative minimum at $x=0$; Relative maximum at $x=2$

(Graph: Plot intercepts $(0,0),(3,0)$; relative min at $(0,0)$, relative max at $(2,\frac{4}{3})$; inflection at $(1,\frac{2}{3})$; curve decreasing then increasing then decreasing, concave up then down.)