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

Step2: Compute first derivative

$y'=-x^2+2x$.

Step3: Find critical points

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

Step4: Determine increasing/decreasing

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

Step5: Find relative extrema

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

Step6: Compute second derivative

$y''=-2x+2$.

Step7: Find inflection point

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

Step8: Determine concavity

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

Answer:

• x-intercepts: $x=0, 3$; y-intercept: $y=0$
• Critical points at $x=0, 2$
• Decreasing: $(-\infty,0)\cup(2,\infty)$; Increasing: $(0,2)$
• Relative min at $(0,0)$; Relative max at $(2,\frac{4}{3})$
• Inflection point at $x=1$
• Concave up: $(-\infty,1)$; Concave down: $(1,\infty)$

(Graph: Plot intercepts $(0,0),(3,0)$, extrema $(0,0),(2,\frac{4}{3})$, inflection point $(1,\frac{2}{3})$, then sketch using concavity and monotonicity.)