the vertex can be found by completing the square:
f(x) = -(x^2 - 50x)
= - (x^2 + 625 - 625 - 50x) (take half the coeff of x, square
= - (x^2 + 625 - 50x ) + 625 it then add and subtract it)
= - (x^2 - 25x - 25x + 625) + 625
= - (x(x- 25) -25(x + 25)) + 625
= - ((x- 25) (x + 25)) + 625
= - ((x + 25)^2 + 625
so the vertex is (25, 625)
in this context this is the 'optimized' area or the largest possible area with the given perimeter (so it is basically a square, 25 x 4 = 100)
