for each problem, find the: x and y intercepts, x-coordinates of the critical points, open intervals where…

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.\n\n1) $y = -\\frac{x^3}{3} + x^2$

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.\n\n1) $y = -\\frac{x^3}{3} + x^2$

Answer

Explanation:

Step1: Find x and y intercepts

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

Step2: Find critical points

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

Step3: Determine intervals of increase/decrease

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

Step4: Find inflection points

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

Step5: Determine concavity

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

Step6: Identify relative extrema

At $x=0$, $y=0$ (relative minimum). At $x=2$, $y = -\frac{8}{3} + 4 = \frac{4}{3}$ (relative maximum).

Answer:

x-intercepts: $(0, 0), (3, 0)$; y-intercept: $(0, 0)$ Critical points x-coordinates: $x=0, x=2$ Increasing: $(0, 2)$; Decreasing: $(-\infty, 0), (2, \infty)$ Inflection point x-coordinate: $x=1$ Concave up: $(-\infty, 1)$; Concave down: $(1, \infty)$ Relative minimum: $(0, 0)$; Relative maximum: $(2, \frac{4}{3})$