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Good choice of material is often a compromise of many factors. For instance, eddy currents forces the use of thinner materials, or materials with higher resistivity. Powder cores require smaller particles which in turn leads to lower permeability. Therefore, the usage of a given material is usually restricted from the viewpoint of highest allowed frequency - usually from the viewpoint of power loss.
On the other hand, it is possible to use such material at lower frequency, but this becomes inefficient both because of technical design and associated costs. It might be cheaper to use material designed for lower frequencies, or such material can have lower losses or higher permeability.
The material can be used at frequencies beyond the limits of major frequency range (e.g. see material Ferrotron 119). But the outcome might be worse than desired and in extreme cases the material might be destroyed if too high frequency is applied (due to the losses and increase in temperature).
Usually, lower frequencies can be applied safely, if the performance of the device is not affected. For instance, ordinary power transformers cannot be operated at frequency below the nominal frequency because the magnetic core might saturate, which could lead to catastrophic damage to the windings. However, such damage would be caused by the performance of the whole device, and not the material.