Apply Fermat's principle to a Concave mirror

[KT KIM]

I am now taking optics class at my school. Fermat principle can be applied on mirror of course.

Then what about Concave mirror? According to the calculus of variation. the optimized path(actual path of the light) should be the shortest path. but in the concave mirror case, it goes through the longest path.

Like above.

How can I apply Fermat's principle on concave mirror?

 

[stevendaryl]

I am now taking optics class at my school. Fermat principle can be applied on mirror of course.

Then what about Concave mirror? According to the calculus of variation. the optimized path(actual path of the light) should be the shortest path. but in the concave mirror case, it goes through the longest path.

Like above.

How can I apply Fermat's principle on concave mirror?

With variational principles, the path is not the absolute minimum (or maximum) it is only a local extremum. What this means is the following: If you shift the path slightly to the dotted path shown in the figure below, by shifting the location where the light ray hits the mirror by $\Delta x$, then the change in the path length will be of order $(\Delta x)^2$. That's a statement about the limit as $\Delta x \rightarrow 0$. Fermat's principle doesn't say anything about paths that are very different from the given path, it only applies to infinitesimally different paths.