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: Compute first derivative

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

Step4: Find critical points

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

Step5: 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)

Step6: Find relative extrema

• $x=0$: change from decreasing to increasing → relative minimum
• $x=2$: change from increasing to decreasing → relative maximum

Step7: Compute second derivative

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

Step8: Find inflection points

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

Step9: Determine concavity intervals

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 at x=0,2; increasing on (0,2), decreasing on (-∞,0)∪(2,∞); relative min at x=0, relative max at x=2; inflection point at x=1; concave up on (-∞,1), concave down on (1,∞).
(Graph: passes through (0,0) and (3,0); has min at (0,0), max at (2, 4/3); inflection at (1, 2/3); concave up left of x=1, concave down right of x=1.)