Question

for each problem, find the: x and y intercepts, x - coordinates of the critical points, open intervals wher the function is increasing and decreasing, x - coordinates of the inflection points, open intervals where tl 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 x-intercepts

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

Step2: Find y-intercept

Set $x=0$: $y=0$

Step3: Find critical points

First derivative: $y' = -x^2 + 2x$
Set $y'=0$: $-x(x-2)=0 \Rightarrow x=0$ or $x=2$

Step4: Determine increasing/decreasing intervals

Test intervals:

• $(-\infty,0)$: $y'(-1)=-3<0$ (decreasing)
• $(0,2)$: $y'(1)=1>0$ (increasing)
• $(2,\infty)$: $y'(3)=-3<0$ (decreasing)

Step5: Find relative extrema

• $x=0$: decreasing→increasing (relative min)
• $x=2$: increasing→decreasing (relative max)

Step6: Find inflection points

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

Step7: Determine concavity

Test intervals:

• $(-\infty,1)$: $y''(0)=2>0$ (concave up)
• $(1,\infty)$: $y''(2)=-2<0$ (concave down)

Answer:

• x-intercepts: $x=0, 3$
• y-intercept: $y=0$
• Critical points: $x=0, 2$
• Increasing interval: $(0,2)$
• Decreasing intervals: $(-\infty,0), (2,\infty)$
• Relative minimum at $x=0$
• Relative maximum at $x=2$
• Inflection point at $x=1$
• Concave up interval: $(-\infty,1)$
• Concave down interval: $(1,\infty)$

(Graph sketch: Plot intercepts (0,0),(3,0); relative min at (0,0), relative max at (2, 4/3); inflection at (1, 2/3); curve decreasing→increasing→decreasing, concave up→concave down at x=1.)