Question

for each problem, find the: x and y intercepts, x - coordinates of the critical points, open intervals whe 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

Compute $y'$: $y' = -x^2 + 2x$
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

Compute $y''$: $y'' = -2x + 2$
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: 0, 3
y-intercept: 0
Critical points x: 0, 2
Increasing: (0,2); Decreasing: (-∞,0)∪(2,∞)
Relative min: (0,0); Relative max: (2, 4/3)
Inflection point x:1
Concave up: (-∞,1); Concave down: (1,∞)
(Sketch uses above info: passes through (0,0),(3,0); has min at (0,0), max at (2,4/3); inflection at x=1, concave up left of 1, down right of 1.)