Chapter 3.5, Problem 28E

To determine

To show:That the graph of $y=\tan x$ and $y=\cot x$ have horizontal tangents at $x=0$ .

Answer to Problem 28E

Both equations have no horizontal tangents becausetheir derivative is never equal to $$

Explanation of Solution

Given information:Given equation is $y=\tan x$ and $y=\cot x$ .

Calculation:Given equation is $y=\tan x$ and $y=\cot x$ .

  $\begin{array}{l}y=\tan x\\ \frac{dy}{dx}={\sec }^{2}x\end{array}$

Which is never equal to $0$ .

Therefore $y=\tan x$ has no horizontal tangent.

  $\begin{array}{l}y=\cot x\\ \frac{dy}{dx}=-\cos e{c}^{2}x\end{array}$

Which is never equal to $0$ .

Therefore $y=\cot x$ has no horizontal tangent.