Question

find two numbers whose sum is 30, such that the sum of a square of one number plus ten times the other number is a minimum

Answer 1

we want to minimize\[x^2 +10y\]subject to \[x+y=30\]so\[y=30-x\]sub into for y\[x^2 +10(30-x)\]\[x^2-10x+300\]now we can minimize the above equation by differentiating and setting it equal to zero
the derivative is\[2x-10\]equating to zero gives\[0=2x-10\]\[x=5\]so one number is 5 and the other is\[y=30-5 =25\] thus the two numbers are 5 and 25

Answer 2

Thanks a bunch!!!