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:

• $(-\infty,0)$: $y'(-1)=-1-2=-3<0$ (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 (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)
• $(1,\infty)$: $y''(2)=-4+2=-2<0$ (concave down)

Answer:

x-intercepts: $x=0, 3$; y-intercept: $y=0$; critical points at $x=0,2$; decreasing on $(-\infty,0)\cup(2,\infty)$, increasing on $(0,2)$; relative min at $x=0$, relative max at $x=2$; inflection point at $x=1$; concave up on $(-\infty,1)$, concave down on $(1,\infty)$; sketch: passes through (0,0),(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.