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 $y=0$: $-\frac{x^3}{3}+x^2=0 \Rightarrow x^2(-\frac{x}{3}+1)=0 \Rightarrow x=0,3$. Set $x=0$: $y=0$.

Step2: Compute first derivative

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

Step3: Find critical points

Set $y'=0$: $-x(x-2)=0 \Rightarrow x=0,2$.

Step4: Determine increasing/decreasing

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

Step5: Identify relative extrema

x=0: min (changes from dec to inc); x=2: max (changes from inc to dec).

Step6: Compute second derivative

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

Step7: Find inflection points

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

Step8: Determine concavity

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

Answer:

x-intercepts: 0, 3; y-intercept: 0; critical points at x=0, 2; increasing on (0,2), decreasing on (-∞,0)∪(2,∞); relative min at x=0, relative max at x=2; inflection point at x=1; concave up on (-∞,1), concave down on (1,∞)