What is the maximum product of two numbers that add to 18 Wh

What is the maximum product of two numbers that add to 18? What numbers yield this product? What is the maximum product of two numbers that add to 26? What numbers yield this product? What is the minimum product of two numbers that differ by 8? What are the numbers? What is the minimum product of two numbers that differ by 7? What are the numbers? What is the maximum product of two numbers that add to - 10? What numbers yield this product? What the maximum product of two numbers that add w - 12? What numbers yield this product?

Solution

17

18. Let the two numbers be x and y.Then x + y = 26 so that y = 26 - x. Then the product P = xy = x ( 26 -x) = -x² + 26x = - (x² -26x) = - (x² - 26x + 169) + 169 = -( x - 13)² + 169. This is a parabola opening downwards with vertex at ( 13, 169). Since the maximum of a parabola opening downwards occurs at the vertex, therefore, P is maximum when x = 13. Then y = 26 - 13 = 13 and the maximum of P is x*y = 13*13 = 169.The numbers that yield this product are 13, 13.

19. Let the two numbers be x and y and let x be the larges of the two numbers. Then x - y = 8 so that y = x - 8.Then the product P = xy = x ( x - 8) = x² -8x = (x² -8x + 16 ) - 16 = ( x - 4)² - 16. This is a parabola opening upwards with vertex at ( 4, -16). Since the minimum of a parabola opening upwards occurs at the vertex, therefore, P is minimum when x = 4. Then y = x - 4 = 4 - 8 = - 4 and the minimum of P is x*y = 4* (-4 )= -16 The numbers that yield this product are 4, - 4.

20.Let the two numbers be x and y and let x be the larges of the two numbers. Then x - y = 7 so that y = x - 7.Then the product P = xy = x ( x - 7) = x² -7x = (x² -2*7/2 x + 49/4 ) - 49/4 = ( x - 7/2)² - 49/4. This is a parabola opening upwards with vertex at ( 7/2, -49/4). Since the minimum of a parabola opening upwards occurs at the vertex, therefore, P is minimum when x = 7/2. Then y = x - 7 = 7/2-7 = -7/2 and the minimum of P is x*y = 7/2*( -7/2) = -49/4 The numbers that yield this product are 7/2, - 7/2.

21. Let the two numbers be x and y. Then x + y = -10 so that y = -10 - x. Then the product P = xy = x ( -10 -x) = - x² -10x = - (x² + 10x) = - (x² + 10x + 25) + 25 = -( x + 5)² + 25. This is a parabola opening downwards with vertex at ( -5, 25). Since the maximum of a parabola opening downwards occurs at the vertex, therefore, P is maximum when x = -5. Then y = -10 - (-5) = -5 and the maximum of P is x*y = (-5)*(-5) = 25.The numbers that yield this product are -5, -5 .

22. Let the two numbers be x and y. Then x + y = -12 so that y = -12 - x. Then the product P = xy = x ( -12 -x) = - x² -12x = - (x² + 12x) = - (x² + 12x + 36) + 36 = -( x + 6)² + 36. This is a parabola opening downwards with vertex at ( -6, 36). Since the maximum of a parabola opening downwards occurs at the vertex, therefore, P is maximum when x = -6. Then y = -12 - (-6) = -6 and the maximum of P is x*y = (-6)*(-6) = 36.The numbers that yield this product are -6, -6 .