# 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.

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.

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

$$-\infty0,\,y">0$$ Slope increasing at decreasing rate. Concave down (and so spills water) right of $$y$$ axis

$$00,\, y'>0,\,y"<0$$

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

• Note the first derivative of $\tanh(x)$ is $\frac{1}{\cosh^2(x)}$. – John Omielan Aug 15 at 2:36
• Thanks for pointing to the typo.. – Narasimham Aug 15 at 3:30