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 $x=0$: $y=0$

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

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 (min, $y=0$)
• $x=2$: increasing→decreasing (max, $y=\frac{4}{3}$)

Step6: Find inflection points

Second derivative: $y''=-2x + 2$
Set $y''=0$: $x=1$ (inflection point, $y=\frac{2}{3}$)

Step7: Determine concavity

Test intervals:

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

Answer:

x-intercepts: 0, 3; y-intercept: 0; critical points: 0, 2; increasing on (0,2), decreasing on (-∞,0)∪(2,∞); relative min at (0,0), relative max at (2, 4/3); inflection point at (1, 2/3); concave up on (-∞,1), concave down on (1,∞)
(Sketch: Plot intercepts (0,0),(3,0), extrema (0,0),(2,4/3), inflection point (1,2/3); curve decreasing then increasing then decreasing, concave up then down.)