Final answer:
The second derivative of the function f(x) = x¶ at x=0 is 0, but this point is not an inflection point because there's no change in concavity around x=0.
Step-by-step explanation:
To show that for f(x) = x⁶ we have f""(0) = 0 but the point x=0 is not an inflection point of f, we must first calculate the second derivative of f. The first derivative of f(x) = x⁶ is f'(x) = 6x⁵, and the second derivative is f"(x) = 30x⁴. Evaluating this at x=0 gives us f"(0) = 30(0)⁴ = 0. However, an inflection point occurs where the second derivative changes sign, which would indicate a change in concavity. Because f"(x) involves an even power of x, it will not change sign around x=0. Thus, x=0 is not an inflection point of f despite the second derivative being zero at that point.