The profit function, y, for selling x shirts is given by the equation y=-0.1x^2+10x. what is the maximum value of the profit, y?

2 ways
1 calcluls
2. algebra

calculus, just take the deritivive and find wher it is 0
y=-0.2x+10
at x=50
sub 50 for x in original
y=250

algebra
for
y=a(x-h)²+k
(h,k) is vertex
k is maximum value

complete the square
y=(-0.1x²+10x)
y=-0.1(x²-100x)
take 1/2 of -100 and square it and add negative and positive inside
-100/2=-50, (-50)²=2500
y=-0.1(x²-100x+2500-2500)
factor perfect square
y=-0.1((x-50)²-2500)
exand
y=-0.1(x-50)²+250

vertex at (50,250)
producing 50 units yeilds a sale price of $250

max price is $250