Question

for each problem, find the: x and y intercepts, x - coordinates of the critical points, open intervals whethe function is increasing and decreasing, x - coordinates of the inflection points, open intervals wherefunction 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: Find critical points

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

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

At $x=0$: decreasing→increasing (minima)
At $x=2$: increasing→decreasing (maxima)

Step6: Find inflection points

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

Step7: Determine concavity

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

Answer:

x-intercepts: 0, 3
y-intercept: 0
Critical points x: 0, 2
Increasing: (0,2); Decreasing: (-∞,0)∪(2,∞)
Relative minima at x=0, relative maxima at x=2
Inflection point x:1
Concave up: (-∞,1); Concave down: (1,∞)
(Sketch: Plot intercepts, extrema, inflection point; curve decreasing then increasing then decreasing, concave up then down)