Nonnegative real numbers a and b satisfy a−b=20. Find the maximum value of a−5b.
Solution — click to reveal
Let x=a and y=b, so x−y=20,a=x2, and b=y2. Then a−5b=x2−5y2=(y+20)2−5y2=−4y2+40y+400=−4(y−5)2+500.The maximum of 500 occurs when y=5, so x=25,a=625, and b=25.