Question 8.7.1: Show that the function f (x) = x^4 does not have an inflecti......

Essential Mathematics for Economic Analysis [1185377]

Show that the function $f (x) = x^{4}$ does not have an inflection point at x = 0, even though $f^{\prime \prime}(0) = 0.$

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Here $f^{\prime }(x) = 4x^{3}$ and $f^{\prime \prime}(x) = 12x^{2},$ so that $f^{\prime \prime} (0) = 0.$ But $f^{\prime \prime}(x) > 0$ for all $x\neq 0$ , and so $f^{\prime \prime}$ does not change sign at x = 0. Hence, x = 0 is not an inflection point—in fact, it is a global minimum, as shown in Fig. 8.6.3.

8.6-3

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