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 intercepts

Set $x=0$: $y=0$. Set $y=0$: $-\frac{x^3}{3}+x^2=0\Rightarrow x^2(3-x)=0\Rightarrow x=0,x=3$.

Step2: Find critical points

First derivative: $y'=-x^2+2x=-x(x-2)$. Set $y'=0$: $x=0,x=2$.

Step3: Determine increasing/decreasing

Test intervals: $(-\infty,0)$: $y'<0$ (decreasing); $(0,2)$: $y'>0$ (increasing); $(2,\infty)$: $y'<0$ (decreasing).

Step4: Find relative extrema

At $x=0$: decreasing→increasing (min); at $x=2$: increasing→decreasing (max).

Step5: Find inflection points

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

Step6: Determine concavity

Test intervals: $(-\infty,1)$: $y''>0$ (up); $(1,\infty)$: $y''<0$ (down).

Answer:

• x-intercepts: $0,3$; y-intercept: $0$
• Critical points: $0,2$
• Increasing: $(0,2)$; Decreasing: $(-\infty,0)\cup(2,\infty)$
• Relative min at $x=0$; Relative max at $x=2$
• Inflection point: $x=1$
• Concave up: $(-\infty,1)$; Concave down: $(1,\infty)$
• (Sketch: Plot intercepts (0,0),(3,0), extrema (0,0),(2,4/3), inflection (1,2/3), then connect with curves matching monotonicity and concavity.)