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

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

Step5: Find relative extrema

$x=0$: min ($y=0$); $x=2$: max ($y=-\frac{8}{3}+4=\frac{4}{3}$).

Step6: Compute second derivative

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

Step7: Find inflection points

Set $y''=0$: $x=1$ (inflection point at $(1,\frac{2}{3})$).

Step8: Determine concavity

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

Answer:

x-intercepts: 0, 3; y-intercept: 0
Critical points x: 0, 2
Increasing: (0,2); Decreasing: (-∞,0)∪(2,∞)
Relative min at (0,0); Relative max at (2, 4/3)
Inflection point x:1
Concave up: (-∞,1); Concave down: (1,∞)
(Graph: Plot intercepts (0,0),(3,0), extrema (0,0),(2,4/3), inflection (1,2/3); connect with curve decreasing then increasing then decreasing, concave up then down.)