Question

question 10 (5 points)
listen
estimate and classify the critical points for the graph of each function.
(0.5, 7), minimum; (2, 1), point of inflection; (3.5, -5), maximum
(0.5, 7), maximum; (2, 1), point of inflection; (3.5, -5), minimum
(0.5, 7), maximum; (3.5, -5), minimum
no critical points

Explanation:

Step1: Recall the definitions

A critical point is a point on the graph of a function where the derivative is zero or undefined. A local maximum is a point where the function changes from increasing to decreasing. A local minimum is a point where the function changes from decreasing to increasing. A point of inflection is a point where the concavity of the function changes.

Step2: Analyze the graph

Looking at the graph, at the point \((0.5,7)\), the function changes from increasing to decreasing, so it is a local maximum. At the point \((3.5, - 5)\), the function changes from decreasing to increasing, so it is a local minimum. The point \((2,1)\) is a point where the concavity of the function changes (from concave - down to concave - up), so it is a point of inflection.

Answer:

\((0.5,7)\), maximum; \((2,1)\), point of inflection; \((3.5, - 5)\), minimum