find 2 positive numbers whose product is a maximum and whose sum of the first number and four times the second number equals 200

Let the numbers be x and y

x+4y = 200

x= 200-4y

Product of the numbers = (200-4y)*y =200y-4y^2

This is maximum when the derivative is zero and second derivative is negative

200-8y=0

8y=200

y=200/8 =25

Second derivative =-8 <0

x= 4*25=100

Product =2500

Answer : x=100, y=25

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