By a *generalized trigonometric polynomial*, I shall mean a function $f:\mathbb R^+ \rightarrow \mathbb R$ given by an expression of the form  
$$f(x) := \sum_{j=1}^k a_j \cos(\alpha_j x) + b_j \sin(\alpha_j x)$$
where $a_j, \alpha_j$ and $b_j$ are arbitrary real numbers such that $\{\alpha_j\}_{j=1}^k$ are all distinct. We assume that $f$ is *non-periodic*. 

I am interested in finding useful absolute lower bounds on $|f'(r)|$ at the roots $r$ of $f$, which are valid uniformly for all sufficiently large zeroes $r$ (and in particular, are independent of $r$ itself). 

Plotting a few trigonometric polynomials suggested that such an absolute lower bound should hold true, however I am not sure how to show such an inequality for $k>1$. I feel like there should be some result in literature on this, however despite an extensive search, I have not found any. Some attempts at using the Cauchy-Schwarz inequality have not led anywhere either. I would really appreciate any suggestions or references. Thank you.

Edit: After Oleg Eroshkin's counterexample, I have modified the question; hopefully, it excludes trivial counterexamples now.