Does Tanh(x) have areas which are concave up or down? I know the point of inflection is at x = 0, however I am struggling with the second derivative test to identify the places where Tanh(x) is concave up or down. 
 A: For the first derivative,
\begin{align}
  \frac{d}{dx} \tanh(x) &= \text{sech}^2(x)\\
  &= \frac{1}{\cosh^2(x)}
\end{align}
Then the second derivative is:
\begin{align}
  \frac{d^2}{dx^2} \tanh(x) &= \frac{d}{dx} \frac{1}{\cosh^2(x)}\\
  &= \frac{-2\sinh(x)}{\cosh^3(x)}
\end{align}
We know that $\cosh(x) \ge 1 > 0\ \forall x \in \mathbb{R}$ (that is, $\cosh$ is always positive and non-negative). $\sinh(x)$ satisfies:
\begin{align}
  \sinh(x) &> 0 \quad \text{if } x > 0\\
  \sinh(x) &= 0 \quad \text{if } x = 0\\
  \sinh(x) &< 0 \quad \text{if } x < 0
\end{align}
Hence,
\begin{align}
  \frac{d^2}{dx^2} \tanh(x) &< 0 \quad \text{if } x > 0\\
  \frac{d^2}{dx^2} \tanh(x) &= 0 \quad \text{if } x = 0\\
  \frac{d^2}{dx^2} \tanh(x) &> 0 \quad \text{if } x < 0
\end{align}
Which tells you exactly when $\tanh$ is concave up and down. I strongly recommend plotting each of $\tanh,\sinh,\cosh$ on wolframalpha.com to get a feel for how they look.
A: Find first and second derivatives 
$$\frac{1}{\cosh^2(x)},\, \frac{-2\sinh(x)}{\cosh^3(x)}; $$
Slope increasing at increasing rate. Concave up (and so holds water) left of $y$ axis 
$$-\infty<x< 0, y<0,\, y'>0,\,y">0 $$ Slope increasing at decreasing rate. Concave down  (and so spills water) right of $y$ axis
$$0<x<+\infty,  y>0,\, y'>0,\,y"<0 $$
At inflection point/origin locally a straight line  $ y^"$ has no sign i.e, sign/magnitude is zero
$$ x=0, \, y'=1,\,y"=0 \,$$
To appreciate the above fully..  plot not just $y=\tanh(x)$ but also $y^{'}, y^{''}.$
