Explanation Given : f ( x 1 , ..... x n ) = x 1 + x 2 + ... + x n , constraints x 1 2 + .... + x n 2 = B 2 Formula Used : ∇ f = λ ∇ g Proof : Step 1: First as we see the function is continuous and the constraint is a sphere centered at the origin in the nth-dimensional space, hence f has extreme values on this set. (For n=2, the constraint defines the circle x 2 + y 2 = B 2 ). We must maximize f ( x 1 , ..... x n ) = x 1 + x 2 + ... + x n subject to the constraint g ( x 1 , ... , x n ) = x 1 2 + .... + x n 2 = B 2 = 0 Step 2: While writing out the Lagrange Equations, the gradient vectors are as follows: ∇ g = 〈 2 x 1 , 2 x 2 , ... , 2 x n 〉 ∇ f = 〈 1 , 1.... , 1 〉 The Lagrange Condition ∇ f = λ ∇ g yields the following equations: 1 = λ ( 2 x 1 ) 1 = λ ( 2 x 1 ) ⋮ 1 = λ ( 2 x n ) Step 3: Solving for λ in terms of x 1 , x 2 .. x n The Lagrange equations imply that x j ≠ 0 for j=1, 2, ..n Therefore, we may divide by x j to obtain λ = 1 2 x 1 λ = 1 2 x 2 ⋮ λ = 1 2 x n Step 4: Solving for x 1 , x 2 .. x n using the constraint. Equating the expressions for λ gives the following equations: 1 2 x 1 = 1 2 x n ⇒ x 1 = x n 1 2 x 2 = 1 2 x n ⇒ x 2 = x n ⋮ 1 2 x n − 1 = 1 2 x n − 1 ⇒ x n − 1 = x n Substituting the constraint x 1 , x 2 , ...

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Chapter 15.8, Problem 54E

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To prove: The maximum of $f (x_{1}, ..... x_{n}) = x_{1} + x_{2} + ... + x_{n}$ subject to the constraints $x_{1}^{2} + .... + x_{n}^{2} = B^{2}$ is $\sqrt{n} B$ and to show that

$| a_{1} | + | a_{2} | ... + | a_{n} | \leq \sqrt{n} {(a_{1}^{2} + .... + .. a_{n}^{2})}^{\frac{1}{2}}$

for all positive numbers $a_{1}, ..., a_{n}$

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Chapter 15.8, Problem 53E

Chapter 15.8, Problem 55E

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Chapter 15 Solutions

CALCULUS (CLOTH)

Ch. 15.1 - Prob. 1PQ Ch. 15.1 - Prob. 2PQ Ch. 15.1 - Prob. 3PQ Ch. 15.1 - Prob. 4PQ Ch. 15.1 - Prob. 5PQ Ch. 15.1 - Prob. 1E Ch. 15.1 - Prob. 2E Ch. 15.1 - Prob. 3E Ch. 15.1 - Prob. 4E Ch. 15.1 - Prob. 5E

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To prove : The maximum of f ( x 1 , ..... x n ) = x 1 + x 2 + ... + x n subject to the constraints x 1 2 + .... + x n 2 = B 2 is n B and to show that | a 1 | + | a 2 | ... + | a n | ≤ n ( a 1 2 + .... + .. a n 2 ) 1 2 for all positive numbers a 1 , ... , a n

Solution Summary: The author explains the Lagrange Equations, where f is a sphere centered at the origin in the nth-dimensional space.

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To prove : The maximum of f ( x 1 , ..... x n ) = x 1 + x 2 + ... + x n subject to the constraints x 1 2 + .... + x n 2 = B 2 is n B and to show that | a 1 | + | a 2 | ... + | a n | ≤ n ( a 1 2 + .... + .. a n 2 ) 1 2 for all positive numbers a 1 , ... , a n

Solution Summary: The author explains the Lagrange Equations, where f is a sphere centered at the origin in the nth-dimensional space.

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To prove: The maximum of f(x1,.....xn)=x1+x2+...+xn subject to the constraints x12+....+xn2=B2 is nB and to show that |a1|+|a2|...+|an|≤n(a12+....+..an2)12 for all positive numbers a1,...,an

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Chapter 15 Solutions