Question 8.7.3: Find possible inflection points for f (x) = x^6 − 10x^4....

In this case f^{\prime}(x)=6x^{5}-40x^{3}{\mathrm{~and}}

A sign diagram for f^{\prime\prime} is as follows:

Figure (1)

From the sign diagram we see that f^{\prime\prime} changes sign at x = −2 and at x = 2, so these are inflection points. Since f^{\prime\prime} does not change sign at x = 0, it is not an inflection point, even though f^{\prime\prime}(0) = 0.