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 intervals

Test $y'$:

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

Step5: Find relative extrema

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

Compute values:

• $y(0)=0$ (min)
• $y(2)=-\frac{8}{3} + 4 = \frac{4}{3}$ (max)

Step6: Find inflection points

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

Step7: Determine concavity intervals

Test $y''$:

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

Answer:

x-intercepts: 0, 3; y-intercept: 0
Critical points: 0, 2
Increasing: (0,2); Decreasing: (-∞,0)∪(2,∞)
Relative min at x=0 (value 0); Relative max at x=2 (value 4/3)
Inflection point: 1
Concave up: (-∞,1); Concave down: (1,∞)
(Graph: Plot intercepts, extrema, inflection point; connect with curves matching increasing/decreasing and concavity)