# Single Variable Calculus, Chapter 4, 4.3, Section 4.3, Problem 64

Prove that if $f(x) = x^4$, then $f''(0) = 0$, but (0,0) is not an inflection point of $f$.

\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}

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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