Question

Find three non-negative numbers a, b, and c such that their product is a maximum when the numbers a, b, and c are constrained by the relation 7abc = 1?

Answer 1

To maximize the product abc with the constraint 7abc = 1, set a = b = c to the cube root of 1/7.

Step-by-step explanation:

To find the non-negative numbers a, b, and c such that their product abc is maximized given the constraint 7abc = 1, we can apply the concept of geometric mean.

First, because 7abc = 1, we can write abc = 1/7. Now, the geometric mean of three numbers is their cube root product, and for a set of numbers, the geometric mean is maximized when all the numbers are equal. Therefore, to maximize abc, we should have a = b = c.

So, we need to solve the equation: a^3 = 1/7. Taking the cube root of both sides, we get: a = b = c = (1/7)^(1/3). That is the value of each variable that maximizes the product abc under the given constraint.