Question

for each problem, find the: x and y intercepts, x - coordinates of the critical points, open intervals wher the function is increasing and decreasing, x - coordinates of the inflection points, open intervals where 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

Test intervals:

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

Step5: Find relative extrema

• $x=0$: decreasing→increasing (minima)
• $x=2$: increasing→decreasing (maxima)

Step6: Find inflection points

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

Step7: Determine concavity

Test intervals:

• $(-\infty,1)$: $y''(0)=2>0$ (concave up)
• $(1,\infty)$: $y''(2)=-2<0$ (concave down)

Answer:

• x-intercepts: $x=0, 3$
• y-intercept: $y=0$
• Critical points: $x=0, 2$
• Increasing interval: $(0,2)$
• Decreasing intervals: $(-\infty,0), (2,\infty)$
• Relative minima at $x=0$
• Relative maxima at $x=2$
• Inflection point: $x=1$
• Concave up: $(-\infty,1)$
• Concave down: $(1,\infty)$

(Graph sketch: Passes through (0,0) and (3,0); has minimum at (0,0), maximum at (2, 4/3); inflection at (1, 2/3); concave up left of x=1, concave down right of x=1.)