we need to find the area with respect of s and r then derive it to find the maximum values

ok

Attachment 14705

I divide the triangle to three equal triangles the hight of the small triangle is

using Pythagorean theorem

the area of every small triangle is

so the area of the large triangle is

critical points

but we have length so ignore negatives

we have two positive critical points study the sign of the Area around these two

the function is not define for s>=2r and s<=-2r

for s values -r sqrt(2) < s< r sqrt(2) the first derivative is positive so the function is increasing

for s> r sqrt(2) take for example 1.5r

for in this interval the first derivative is negative the function is decreasing

so you have s= r sqrt(2) as a maximum point

is there something you can't understand