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 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 $x=0$: $y=-\frac{0^3}{3} + 0^2 = 0$

Step3: Find critical points

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

Step4: Determine increasing/decreasing

Test intervals:

• $(-\infty,0)$: $y'(-1)=-1-2=-3<0$ (decreasing)
• $(0,2)$: $y'(1)=-1+2=1>0$ (increasing)
• $(2,\infty)$: $y'(3)=-9+6=-3<0$ (decreasing)

Step5: Find relative extrema

• $x=0$: decreasing→increasing (minima)
• $x=2$: increasing→decreasing (maxima)

Step6: Find inflection points

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

Step7: Determine concavity

Test intervals:

• $(-\infty,1)$: $y''(0)=2>0$ (concave up)
• $(1,\infty)$: $y''(2)=-4+2=-2<0$ (concave down)

Answer:

x-intercepts: 0, 3
y-intercept: 0
Critical points x: 0, 2
Increasing: (0,2); Decreasing: (-∞,0)∪(2,∞)
Relative minima at x=0, relative maxima at x=2
Inflection point x:1
Concave up: (-∞,1); Concave down: (1,∞)
(Sketch: Plot intercepts (0,0),(3,0); minima (0,0); maxima (2, 4/3); inflection (1, 2/3); curve decreasing left of 0, increasing 0-2, decreasing right of 2; concave up left of 1, concave down right of 1)