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

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 intercepts

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

Step2: Find critical points

Differentiate $y = -\frac{x^3}{3} + x^2$ and set to zero. $$y' = -x^2 + 2x = x(2 - x) = 0 \Rightarrow x = 0, x = 2$$

Step3: Determine monotonicity

Test intervals $(-\infty, 0)$, $(0, 2)$, and $(2, \infty)$ in $y'$. $$y' < 0 \text{ on } (-\infty, 0) \cup (2, \infty); y' > 0 \text{ on } (0, 2)$$

Step4: Identify extrema

Evaluate $y$ at critical points. $y(0) = 0$, $y(2) = -\frac{8}{3} + 4 = \frac{4}{3}$. $$\text{Min: } (0, 0), \text{ Max: } (2, \frac{4}{3})$$

Step5: Find inflection points

Differentiate $y'$ and set to zero. $$y'' = -2x + 2 = 0 \Rightarrow x = 1$$

Step6: Determine concavity

Test intervals $(-\infty, 1)$ and $(1, \infty)$ in $y''$. $$y'' > 0 \text{ on } (-\infty, 1) \text{ (Up)}; y'' < 0 \text{ on } (1, \infty) \text{ (Down)}$$

Answer:

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