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

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

Answer

Explanation:

Step1: Find x-intercepts (y=0)

Set ( y = 0 ): ( -\frac{x^3}{3}+x^2 = 0 )
Factor: ( x^2\left(-\frac{x}{3}+1\right)=0 )
Solutions: ( x^2 = 0 ) or ( -\frac{x}{3}+1 = 0 )
( x = 0 ) (double root) or ( x = 3 )

Step2: Find y-intercept (x=0)

Substitute ( x = 0 ) into ( y ): ( y = -\frac{0^3}{3}+0^2 = 0 )

Step3: Find critical points (first derivative = 0)

First derivative: ( y' = -x^2 + 2x )
Set ( y' = 0 ): ( -x^2 + [LLM SSE On Failure]