Answer :
Final answer:
The local minimum of the function y = -x^6 - 6x^5 + 50x^3 + 45x^2 - 108x -108 can be found by computing its derivative and solving for 'x' when the derivative equals zero. The resulting x-coordinates can be plugged back into the original function to find the y-values of the local minima. The provided options should be verified by checking the behavior of the derivative around these points.
Explanation:
To find the local minimum of a polynomial function, you need to first compute its derivative, set the derivative equal to zero, and solve for 'x'. The local minimums occur where the derivative of a function changes from negative to positive.
In this case, the derivative of the function y = -x^6 - 6x^5 + 50x^3 + 45x^2 - 108x - 108 is:
-6x^5 - 30x^4 + 150x^2 + 90x - 108
Setting this equal to zero and solving for 'x' yields a complex polynomial equation, which would need to be solved via numerical or analytical methods. The solutions to this equation will give the x-coordinates of the local minima. To find the corresponding y-coordinates, we simply plug back these x-coordinates into the original function y.
The provided options A. (-3,0), B. There is no local minimum, C. (0.618, -146.353), and D. (0, -108) can be checked by computing the derivative at these points and checking the sign change. Option C. (0.618, -146.353) could be the potential local minimum point depending on the behavior of the function around this point.
Learn more about Local Minimum here:
https://brainly.com/question/34256226
#SPJ11
