Why does light take an extremum path according to Fermat's Principle?

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Fermat's Principle states that light travels along paths that minimize or maximize travel time, leading to the concept of extremum paths. The physical reasoning behind this behavior is tied to the wave nature of light and its interactions with different media. While light typically takes the shortest path, it can also take longer routes under certain conditions, such as when reflecting or refracting. The discussion emphasizes that the extremum principle is an inherent characteristic of light's behavior, regardless of the specific path lengths involved. Ultimately, this principle illustrates the complex nature of light's travel and its dependence on the surrounding environment.
Nitin Gupta
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According to Fermat's Principle (modified version) a light ray takes an extremum path. Can anyone explain the physical reason behind this? I mean, sure the light can take the least path but how can it take the longest path? And if it can indeed take the longest path, why not the paths shorter than longest? (Please, I don't want mathematical explanation. I am new here, if this is posted at wrong place or any other problem please PM me)
 
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