The turning point of a cubic function y=x^3-6x^2+12x-5 occurs at the point where the derivative of the function is equal to zero. This point represents a critical point where the function changes direction, from increasing to decreasing or vice versa.
In this case, the derivative of the function y=x^3-6x^2+12x-5 is given by y'=3x^2-12x+12. Setting this derivative equal to zero and solving for x, we find the turning point to be x=2.
At x=2, the function reaches a local minimum or maximum. We can determine the nature of this turning point by assessing the second derivative of the function at that point. The second derivative of y=x^3-6x^2+12x-5 is y''=6x-12. Evaluating this second derivative at x=2, we get y''=6(2)-12=0.
Since the second derivative is equal to zero, we cannot conclusively determine whether the turning point at x=2 is a local minimum, local maximum, or inflection point. Further analysis is needed to determine the behavior of the function near this turning point.
Describe the nature of the turning point y=X^3-6x^2+12x-5
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