Finding the inflection factor for a function

While learning more about restrictions and also by-products, I came accross the adhering to trouble on among Stewart is workouts publication. I've been attempting to cover my head around it yet I have not obtained anywhere valuable:

  1. $ \forall x f''(x) $ exists
  2. $ \exists c \in \mathbb{R} $/ $\forall x \neq c, f'(x) > 0 \wedge f'(c) = 0 $
  3. Then, $(c, f(c))$ is an inflection factor .

From this I can collect that:

  • $f$ is continual
  • $f$ is raising $\forall x \neq c$.

My instinct informs me that $(3)$ is incorrect given that I could be able ahead up with a function specified by components that negates the declaration, yet I have not located a means to confirm this. Any kind of reminders would certainly be substantially valued.

2019-05-13 02:37:39
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Answers: 1

Because the 2nd acquired exists almost everywhere, you additionally recognize that $f'$ is continual almost everywhere. Currently think of the function $f'(x)$ ; it declares to the left and also to the right of $c$, and also is $0$ at $c$, and also it is continual. Does that inform you something concerning $f''$, and also therefore concerning concavity?

2019-05-17 11:53:33