Does this function have a (global) minimum?

Question

A good day to everyone.

Does the following function have a (global) minimum:

$$1 + \frac{1}{x} + {\left(1 + \frac{1}{x}\right)}^\theta,~~x\in\mathbb R$$

where

$$\theta = {\displaystyle\frac{3\log 2 - \log 5}{2(\log 5 - 2\log 2)}} > 1?$$

WolframAlpha says it has none.

Answer 1

Lets do some analysis and see if we can convince ourselves that WA is correct.

• A plot of the function (be careful with the imaginary values for $x \lt 0$), shows:

• We find that we have a limit of $2$ as $x \rightarrow \pm \infty$ (of course, we also see this in the plot), but this does not help us. This is our infimum.

• Taking the derivative yields:

$$\displaystyle f'(x) = -\frac{x \ln(8/5) (1/x+1)^{(\ln(8/5)/\ln(25/16))}+2 x \ln(5/4)+2 \ln(5/4)}{2 x^2 (x+1) \ln(5/4)}$$

• If we plot $f'(x)$, we have:

• The limit of the derivative is zero as $x \rightarrow \pm \infty$, but this does not help us.

• Analytically or numerically, we can find no value of $x$ where $f'(x) = 0$, that is, we can find no critical points.

• Conclusion, there is no global minimum (or maximum or local ones either).