Solve the approximate intervals on which the function is y= (3x)/(2e^x+e^-x)

Question

Use graphing and analytical techniques to find the approximate intervals on which the function is:

(a) Increasing

(b) Decreasing

(.c) concave up

(d) concave down

(e) local extreme values

(f) inflection points

My biggest trouble is finding the derivatives. I have two or three different answers for analytical techniques. I am also confused on how to solve this using graph techniques.

Answer 1

This will be difficult to solve without careful graphing.

There is an obvious 0 at x = 0.

The derivative has 2 zeros near +1 and near -5/4 but they are difficult to compute even approximately.

The graph is asymptotic to 0 as x grows large in either direction.

With that information, you should be able to answer the questions.

Oh, writing the derivative...I'll try to write it here

Let z = 2e^x + e^-x

Then the function is 3x/z and the derivative is

(3z - 3x(2e^x - e^-x)/z²