Does Tanh(x) have areas which are concave up or down?

Question

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.

Answer 1

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.

Answer 2

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 $y^{'}, y^{''}.$