HELP calculus optimization problem: fitting thin rod through corridor

[sickle]

I am having trouble conceptualizing a calculus optimization problem.
I can find the answer to the problem by using the procedure but i am quite uncertain of how the equations match up with what's actually going on in the situation!

Problem: What is the max length of widthless rigid pole that can be carried around a corner of two corridors of width a and b meeting at a right angle?

The solution is identical to finding the shortest length of a ladder from the ground to a wall if there's a block in front and blocking the wall.
This 2nd Q. makes sense because we are minimizing the length of the ladder and indeed the math spits out a local min value.

Now why is it that the conditions to finding the longest length is identical to find the shortest ladder??

 

[lanedance]

they are solve for either the local maxima, or local minima of a function, both of which correspond to f'(x)=0, so the solution method is the same.

You could use the 2nd derivative method to check whether it is a max or min, though it is often obvious based on the problem framing

 

[Office_Shredder]

It's because the problems are physically very similar. As you turn around the corner, you want to find a ladder which touches both walls only once (if it touches both walls for an extended period of time, then it really passed through the walls at some point and that's not allowed). If you think about the corner you are trying to navigate as the block and the outside walls of the corner as the wall and floor, you're looking for a ladder which is only capable of touching the floor and wall in one position, not many positions. This is going to be the shortest ladder capable of touching the wall and the floor at the same time.

The more pertinent question though, is why do you think the second question makes sense because it's asking for a minimum, and the first question doesn't because it's asking for a maximum?