Question

Find the local maximum and minimum values and saddle points of the function y² - 6y cos(x)?

Answer 1

To find the local maximum and minimum values and saddle points of the function y² - 6y cos(x), follow these steps: find the critical points, analyze the second derivative, set the second derivative equal to zero to find the inflection points, use the second derivative test to classify the critical points, and finally, plug in the critical and inflection points into the original function.

Step-by-step explanation:

To find the local maximum and minimum values and saddle points of the function y² - 6y cos(x), we need to find the critical points and analyze the second derivative test.

1. Find the critical points by setting the first derivative equal to zero: 2y - 6 cos(x) = 0. Solve for y to get y = 3 cos(x).
2. Take the second derivative: (d²y/dx²) = -6 sin(x).
3. Set the second derivative equal to zero and solve for x to find the points of possible inflection (saddle points): -6sin(x) = 0.
4. Use the second derivative test to classify the critical points as local maximum, local minimum, or neither.
5. Plug the critical points and inflection points into the original function to find their values.