Chapter 5: Exercises with Solutions

Chapter 5Quadratic Functionsy = − b2a = − 122(1) = −6,and x = 12 + y = 12 + (−6) = 6. So the two numbers are 6 and −6.25. Let x and y be the two numbers, so y = 2x + 3. The goal is to minimize theproduct P = xy, but we cannot minimize a function of more than one variable. Wemust substitute one of the variables out. Substitute y = 2x + 3 into P to get...Simplifying yieldsP (x) = x(2x + 3)P (x) = 2x 2 + 3xIt is an upward parabola, so the minimum occurs atx = − b2a = − 32(2) = −3 4 .It follows that y = 2x + 3 = 2(−3/4) + 3 = −3/2 + 3 = 3/2. So, the two numbers are−3/4 and 3/2.27. Let x and y be the two numbers, so x + y = −10. The goal is to minimize thesum of squares S = x 2 + y 2 . Since y = −10 − x, S can be rewritten as a function of x:Simplifying yieldsS(x) = x 2 + (−10 − x) 2S(x) = 2x 2 + 20x + 100The minimum occurs atand y = −10 − −5 = −5.x = − b2a = −5,29. Let x and y be the two numbers, so x + y = 14. The goal is to minimize the sumof squares S = x 2 + y 2 . Since y = 14 − x, S can be rewritten as a function of x:Simplifying yieldsS(x) = x 2 + (14 − x) 2S(x) = 2x 2 − 28x + 196The minimum occurs atx = − b2a = 7,Version: Fall 2007