Andrew has a favorite function \(A(x)=px+q^x\) such that \(A(1)=4\) and \(4A(2)=37\), find the maximum value of \(p-q.\)
+128475 A(1) = p(1) + q^1 = p + q = 4 ⇒ q = 4 - p (1)
4A(2) = 4 [ p(2) + q^2 ] = 8p + 4q^2 = 37 (2)
Sub (1) into (2) for q
8p + 4 (4 - p)^2 = 37
8p + 4 (p^2 - 8p +16) = 37
8p + 4p^2 - 32p + 64 = 37
4p^2 - 24p + 27 = 0 factor
(2p - 9)(2p - 3) = 0
Set each factor to 0 and solve for p and we get that p = 9/2 or p =3/2
When p = 9/2, q = 4 - (9/2) = -1/2 and p - q = 9/2 - - 1/2 = 10/2 = 5
When p= 3/2 , q = 4 - (3/2) = 5/2 and p - q = 3/2 - 5/2 = -2/2 = - 1
So max value of p - q = 5
