Explanation Result used: The vertex and the y -intercept of the quadratic function y = a x 2 + b x + c is ( − b 2 a , f ( − b 2 a ) ) and ( 0 , c ) . Description: In general, the equation of the parabola is y = a x 2 + b x + c . Substitute x = − 1 and y = 14 in y = a x 2 + b x + c . 14 = a ( − 1 ) 2 + b ( − 1 ) + c 14 = a − b + c 14 + b = a + c Similarly substitute x = 1 and y = 9 in y = a x 2 + b x + c . 9 = a ( 1 ) 2 + b ( 1 ) + c 9 = a + b + c 9 − b = a + c On further simplification, 9 − b = 14 + b [ ∵ a + c = 9 − b ] 2 b = − 5 b = − 5 2 So, the value of b is − 5 2 . Substitute b = − 5 2 in 9 − b = a + c . 9 + ( 5 2 ) = a + c 23 2 = a + c ( 1 ) Substitute x = 2 and y = 8 in y = a x 2 + b x + c

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Maximization

Have you ever seen a bridge in the shape of a parabola? Where is its maximum on the bridge? The maximum of that bridge is the highest point on that bridge. The process of finding the maximum of the bridge here is called the maximization.

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Chapter 7.4, Problem 38E

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The equation of the parabola that passes through the points (−1,14), (1,9) and (2,8).

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Chapter 7.4, Problem 37E

Chapter 7.4, Problem 39E

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Suppose we have a linear program in standard equation form maximize c'x subject to Ax=b, x≥ 0. and suppose u, v, and w are all optimal solutions to this linear program. (a) Prove that zu+v+w is an optimal solution. (b) If you try to adapt your proof from part (a) to prove that that u+v+w is an optimal solution, say exactly which part(s) of the proof go wrong. (c) If you try to adapt your proof from part (a) to prove that u+v-w is an optimal solution, say exactly which part(s) of the proof go wrong.

(a) For the following linear programme, sketch the feasible region and the direction of the objective function. Use you sketch to find an optimal solution to the program. State the optimal solution and give the objective value for this solution. maximize +22 subject to 1 + 2x2 ≤ 4, 1 +3x2 ≤ 12, x1, x2 ≥0 (b) For the following linear programme, sketch the feasible region and the direction of the objective function. Explain, making reference to your sketch, why this linear programme is unbounded. maximize ₁+%2 subject to -2x1 + x2 ≤ 4, x1 - 2x2 ≤4, x1 + x2 ≥ 7, x1,x20 Give any feasible solution to the linear programme for which the objective value is 40 (you do not need to justify your answer).

find the domain of the function f(x)

Chapter 7 Solutions

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Ch. 7.1 - Solve for x: x2 + 4x − 21 = 0 Ch. 7.1 - Prob. 2PE Ch. 7.1 - Prob. 1E Ch. 7.1 - Prob. 2E Ch. 7.1 - Prob. 3E Ch. 7.1 - Prob. 4E Ch. 7.1 - Prob. 5E Ch. 7.1 - Prob. 6E Ch. 7.1 - Prob. 7E Ch. 7.1 - Prob. 8E

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The equation of the parabola that passes through the points ( − 1 , 14 ) , ( 1 , 9 ) and ( 2 , 8 ) .

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The equation of the parabola that passes through the points (−1,14), (1,9) and (2,8).

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Chapter 7 Solutions