What is the maximum value of $\frac{2x}{x + 1} + \frac{x}{x - 1}$, if $x \in \mathbb{R}$ and $x > 1$?

What is the maximum value of $$f(x) = \frac{2x}{x + 1} + \frac{x}{x - 1},$$ if $x \in \mathbb{R}$ and $x > 1$?

A 2-D plot of of $f$ for $x \in (\infty, \infty)$ is here.

Lastly, note that WolframAlpha cannot find a global maximum.

Solution 1:

Hint: Use the derivative test. If you differentiate the function and equate to zero you get the two critical points

$$ 3+2\sqrt{2},\, 3-2\sqrt{2}. $$

Now, find the second derivative to test the critical points.

For $x>1$ we have $f(x)>\frac1{x-1}$, which is unbounded from above.

HINT:

Let $$y=\frac{2x}{x+1}+\frac x{x-1}$$

$$\implies x^2(3-y)-x+y=0$$

which is a quadratic equation in $x$

For real $x,$ the discriminant must be $\ge0$