   Chapter 7.6, Problem 21E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Finding Positive Numbers In Exercises 19-22, find three positive numbers x, y, and z that satisfy the given conditions.The sum is 120 and the sum of the squares is a minimum.

To determine

To calculate: The three positive numbers x,yandz that satisfy this statement “The sum is 120 and sum of the square is minimum”.

Explanation

Given Information:

The provided statement “The sum is 120 and sum of the square is minimum”.

Formula used:

Lagrange Multiplier:

F(x,y,z)=f(x,y,z)λg(x,y,z)

Where f(x,y,z)andλg(x,y,z) are the function and the constraint and λ is the Lagrange Multiplier.

Step 1: Consider the number to be x,yandz.

Step 2: Write the function and constraint in the expansion.

Step 3: Partially differentiate f and g with respect to x,yandz.

Step 4: Use the primary formula, find the value of λ.

Step 5: Using λ find the relation between the numbers and arrive at the final form.

Step 6: Substitute the values in the g to find all the three numbers.

Calculation:

Consider x,y,z be the three positive numbers.

Now minimize f(x)=x2+y2+z2 subject to g(x,y,z)=x+y+z=120.

The gradients of f and g are found by solving the system of equations

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