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: 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:

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

Step6: Find relative extrema

• $x=0$: decreasing→increasing (relative min, $y=0$)
• $x=2$: increasing→decreasing (relative max, $y=-\frac{8}{3}+4=\frac{4}{3}$)

Step7: Compute second derivative

$y'' = -2x + 2$

Step8: Find inflection points

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

Step9: 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: $x=0,2$
• Increasing: $(0,2)$; Decreasing: $(-\infty,0)\cup(2,\infty)$
• Inflection point: $x=1$
• Concave up: $(-\infty,1)$; Concave down: $(1,\infty)$
• Relative min: $(0,0)$; Relative max: $(2,\frac{4}{3})$

(Graph: Plot intercepts, extrema, inflection point; sketch using concavity/increase/decrease info)