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

• For $x<0$: $y' < 0$ (decreasing)
• For $0 0$ (increasing)
• For $x>2$: $y' < 0$ (decreasing)

Step6: Identify relative extrema

• $x=0$: decreasing to increasing → relative minimum ($y=0$)
• $x=2$: increasing to decreasing → relative maximum ($y=-\frac{8}{3} + 4 = \frac{4}{3}$)

Step7: Compute second derivative

$y'' = -2x + 2$

Step8: Find inflection points

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

Step9: Determine concavity intervals

• For $x<1$: $y'' > 0$ (concave up)
• For $x>1$: $y'' < 0$ (concave down)

Answer:

• x-intercepts: $x=0, 3$; y-intercept: $y=0$
• Critical points: $x=0, 2$
• Increasing: $(0,2)$; Decreasing: $(-\infty,0)\cup(2,\infty)$
• Relative minimum: $(0,0)$; Relative maximum: $(2,\frac{4}{3})$
• Inflection point: $x=1$
• Concave up: $(-\infty,1)$; Concave down: $(1,\infty)$

(Graph sketch: Plot intercepts, extrema, inflection point; draw curve decreasing then increasing then decreasing, concave up then down.)