Question

find the smallest positive in which lies the following tanx+tanhx=0

Answer 1

We have to find least positive integer interval in which the solution of tanx+tanhx=0 lies.

Let $f(x) = tan(x) + tanh(x)$ .

Now, $f(1) = tan(1) + tanh(1) = 2.319 > 0$ ,

$f(2) = tan(2) + tanh(2) = -1.221 < 0$ ,

and $f(3) = tan(3)+tanh(3) = 0.8525 > 0$ .

We known that $tanh(x) \in (-1,1)$ for all values of $x$ .

Also, $tan(x)$ has infinite discontinuity at $x = \frac{\pi}{2} = 1.57, \frac{3\pi}{2} = 4.71, ...$ .

Hence, least positive integer interval in which the solution of tanx+tanhx=0 lies is $(2,3)$ .