Let $g$ be a function such that all its derivatives exist. We say $g$ has an inflection point at $x_0$ if the second derivative $g''$ changes sign at $x_0$ i.e., if $g''(x_0 - \epsilon) \times g''(x_0 + \epsilon) < 0$ for all small enough positive $\epsilon$. (a) If $g''(x_0) = 0$ then $g$ has an inflection point at $x_0$. True or False? (b) If $g$ has an inflection point at $x_0$ then $g''(x_0) = 0$. True or False? (c) Find all values $x_0$ at which $x^{4}(x - 10)$ has an inflection point.
Let $g$ be a function such that all its derivatives exist. We say $g$ has an inflection point at $x_0$ if the second derivative $g''$ changes sign at $x_0$ i.e., if $g''(x_0 - \epsilon) \times g''(x_0 + \epsilon) < 0$ for all small enough positive $\epsilon$.\\
(a) If $g''(x_0) = 0$ then $g$ has an inflection point at $x_0$. True or False?\\
(b) If $g$ has an inflection point at $x_0$ then $g''(x_0) = 0$. True or False?\\
(c) Find all values $x_0$ at which $x^{4}(x - 10)$ has an inflection point.