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

• $x=0$: $y'$ changes from - to + → relative min
• $x=2$: $y'$ changes from + to - → relative max

Compute values: $y(0)=0$, $y(2)=-\frac{8}{3}+4=\frac{4}{3}$

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)

Step8: Sketch the graph

Plot intercepts (0,0),(3,0), extrema (0,0),(2,4/3), inflection point (1,2/3); use intervals for shape.

Answer:

x-intercepts: 0, 3; y-intercept: 0; critical points at x=0,2; decreasing on $(-\infty,0)\cup(2,\infty)$, increasing on (0,2); relative min at (0,0), relative max at (2,4/3); inflection point at x=1; concave up on $(-\infty,1)$, concave down on (1,$\infty$).
(Graph: Plot key points and follow interval behaviors as above.)