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 $x=0$: $y=-\frac{0^3}{3} + 0^2 = 0$

Step3: Find critical points

Compute $y'$: $y' = -x^2 + 2x$
Set $y'=0$: $-x^2 + 2x = 0 \Rightarrow x(-x + 2) = 0 \Rightarrow x=0$ or $x=2$

Step4: Determine increasing/decreasing intervals

Test $y'$ sign:

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

Step5: Find relative extrema

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

Step6: Find inflection points

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

Step7: Determine concavity intervals

Test $y''$ sign:

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

Answer:

x-intercepts: 0, 3
y-intercept: 0
Critical points x-coordinates: 0, 2
Increasing interval: (0, 2)
Decreasing intervals: (-∞, 0), (2, ∞)
Relative minimum at x=0, relative maximum at x=2
Inflection point x-coordinate: 1
Concave up interval: (-∞, 1)
Concave down interval: (1, ∞)

(Graph sketch: Passes through (0,0) and (3,0); decreases from left to x=0, increases to x=2, then decreases; concave up left of x=1, concave down right of x=1.)