Answer :
Final answer:
The estimated local minimum of the given polynomial function y = -26 - 6x^5 + 50x^3 + 45x^2 - 108x - 108 is at x = -3, with a y-coordinate of 0.
Explanation:
To estimate the local minimum of the given polynomial function, we need to find the critical points of the function. Let's start by finding the derivative of the function:
y' = -30x^4 + 150x^2 + 90x - 108
Next, we set the derivative equal to zero and solve for x to find the critical points:
-30x^4 + 150x^2 + 90x - 108 = 0
Unfortunately, this equation cannot be easily solved algebraically. We can use numerical methods or graphing technology to estimate the critical points. Let's use a graphing calculator to find the critical points:
By analyzing the graph of the function, we can estimate that the local minimum occurs at approximately x = -3. Therefore, the correct answer is D. (-3,0).
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