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: Compute first derivative

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

Step4: Find critical points

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

Step5: Intervals of increase/decrease

Test $y'$:

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

Step6: Relative extrema

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

Step7: Compute second derivative

$y'' = -2x + 2$

Step8: Find inflection points

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

Step9: 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,∞); inflection point: 1; concave up: (-∞,1); concave down: (1,∞); relative min at x=0, relative max at x=2.
(Graph: Plot intercepts (0,0),(3,0); relative min (0,0), relative max (2,4/3); inflection point (1,2/3); sketch curve with concavity changes.)