This really belongs in the "advance applied math section"

For question 1. you should know that themaximumvalue that a friction force is where R is the normal reaction force between the object and the surface it in in contact with and is the coefficient of friction.

(if you don't follow what I say here ask me to post a diagram)

the normal reaction force is given by . therefore the maximum friction force is . If there is to be motion the forward must be greater than the maximum friction force (such that there is a net resultant force forwards).

so we get

which rearranges to

this can rearrange to give

to find the max/min or F you differentiate using the chain rule. to get

we require

which rearranges to

using some simple trig you get and

put this into the formula (with your value of ) I get the minimum value of .

you should be able to notice that F will be a maximum when in which case

Edit: F is actually maximum in the case when in this case you would just be pulling the sledge vertically