Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.SolutionWe have to maxims x2y5 and we have x+y=35
So put x=35−y in x2y5
we get (1225+y2−70y)(y5)
f(y)=1225y5+y7−70y6 ′
(y)=6125y 4
+7y 6
−420y 5
Putting this to zero we get
y=35,y=25
f ′′
(y)=24500y 3
+42y 5
−2100y 4
f ′′
(35)=24500×35 3
+42×35 5
−2100×35 4
f ′′
(35)>0
At y=25
f ′′
(25)=24500×25 3
+42×25 5
−2100×25 4
f ′′
(25)<0
So given function will have maxima when y=25
So x=10
