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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.

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.

Edit 2: If nothing else, it would also be quite interesting to know something about a lower bound on the average size of the absolute derivative of $f$ at the zeroes, something like a lower bound on $$\liminf_{X \rightarrow \infty} \frac1X \sum_{x_n \leq X} |f'(x_n)|^c$$ for some fixed $c>1$, where $x_1 < x_2 < \cdots$ are the zeroes of $f$ in ascending order.

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 for the sake of non-triviality that $f$ is non-constant). 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.

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.

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 for the sake of non-triviality that $f$ is non-constant). I am interested in finding useful absolute lower bounds on $|f'(x_0)|$ at the roots $x_0$ of $f$, which are valid uniformly for all $x_0$ (and in particular, are independent of $x_0$ 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.

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 for the sake of non-triviality that $f$ is non-constant). 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.