 Solution

Use reproducible random starts and retain the best successful result:

The estimate is $x_{\min }\simeq -0.309213$, $f_{\min }\simeq 0.0192409$. This random-restart optimization improves coverage of the different attraction regions, but their number alone does not guarantee global optimality.

Here an analytic check is possible. For $|x|\ge \pi /5$, the envelope gives $f(x)\ge 1-e^{-{\pi }^2/125}\simeq 0.0759$, above the candidate. On $[0,\pi /5]$, $f\ge 1$. In the remaining interval $[-\pi /5,0]$, the derivative condition is equivalent, with $z=-x$, to $\cot (5z)=0.08z$. The left side decreases strictly from $+\infty$ to $-\infty$ while the right side increases, giving one minimum. This certifies the selected well as global rather than relying only on the restart experiment.