Answer :
Final answer:
Determine the local minimum of a given polynomial function by first finding its derivative, then solving for when the derivative equals zero to find potential minimums and maximums. Evaluate the second derivative at these points to differentiate between the two. Use a graphing calculator or other similar tool for solving.
Explanation:
This is a Mathematics problem based on calculus related to finding the local minimum of a polynomial function. The given function is y = -x^6 - 6x^5 + 50x^3 + 45x^2 - 108x - 108. To find the local minimum, we need to take the derivative of this function and set it equal to zero to find the critical points. The critical points are the potential local maximums and minimums of the function.
However, the derivative of this function is complex and may be difficult to solve for zero without the use of a graphing calculator or computer algebra system. If the student has access to such tools, they could use them to solve for the derivative equal to zero and then evaluate the second derivative at those points to determine if they're local minimums (second derivative > 0) or local maximums (second derivative < 0).
Without specific solutions to compare against, it's uncertain which of the provided options would be the correct answer. Option [b], 'there is no local minimum', could be correct if the function always decreases or increases without any turning points. If the function does have turning points, then one of the other options would be right, depending on the specific values of those turning points.
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