Final answer:
To find the maximum and minimum points of the function y=(1)/(3)x^3-(1)/(2)x^2-2x+2, calculate the critical points by setting the first derivative equal to zero, then determine their nature with the second derivative, and evaluate the original function at these points.
Step-by-step explanation:
To determine the maximum and minimum points of the function y=(1)/(3)x3-(1)/(2)x2-2x+2, we need to find the critical points where the first derivative y' is zero or undefined, and then use the second derivative to test for concavity and confirm whether these points are maxima or minima.
First, we differentiate the function to find y':
Then, we set y' to zero and solve for x:
Solving this quadratic equation gives us the potential critical points. To identify if these points are maximum or minimum, we need to evaluate the second derivative, y'', at these x-values:
If y'' > 0 at a critical point, the point is a minimum; if y'' < 0, the point is a maximum.
After finding the critical points and determining their nature, we plug these x-values back into the original function to find the corresponding y-values and thereby locate the exact maximum and minimum points on the graph.