Single Variable Calculus, Chapter 4, 4.3, Section 4.3, Problem 64
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Prove that if $f(x) = x^4$, then $f''(0) = 0$, but (0,0) is not an inflection point of $f$.
$ \begin{equation} \begin{aligned} \text{if } f(x) &= x^4, \text{ then,}\\ \\ f'(x) &= 4x^3\\ \\ f''(x) &= 12x^2 \\ \\ \\ \text{when } x &= 0,\\ \\ f''(0) &= 12(0)^2\\ \\ f''(0) &= 0 \end{aligned} \end{equation} $
However, for any values of $x$ except $x = 0$, $f''(x) = 12x^2 > 0$ or the graph always has upward concavity. It shows that (0,0) is not an inflection point.
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