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:

Answer:

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