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: Increasing/decreasing intervals

Test $y'$:

• $x<0$: $y'<0$ (decreasing)
• $00$ (increasing)
• $x>2$: $y'<0$ (decreasing)

Step5: Relative extrema

• $x=0$: decreasing→increasing (min)
• $x=2$: increasing→decreasing (max)

Step6: Find inflection points

Second derivative: $y'' = -2x + 2$
Set $y''=0$: $x=1$

Step7: Concavity intervals

Test $y''$:

• $x<1$: $y''>0$ (concave up)
• $x>1$: $y''<0$ (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 min at $x=0$; Relative max at $x=2$
Inflection point: $x=1$
Concave up: $(-\infty,1)$; Concave down: $(1,\infty)$
(Sketch: Plot intercepts, extrema, inflection point; curve decreasing→increasing→decreasing, concave up→down at $x=1$)