Question

The product of two numbers is 64.

Minimize the sum of their cubes.

Please continue with any specific instructions or questions related to finding this minimum value.

Answer 1

The minimum value of the sum of the cubes of two numbers whose product is 64 occurs when the two numbers are equal to the cube root of 64, which is 4. Therefore, the minimum sum is 4³ + 4³ = 128.

Step-by-step explanation:

To minimize the sum of the cubes of two numbers, we can leverage the information that their product is 64. Let the two numbers be (x) and (y). According to the given condition, xy = 64 We aim to minimize the expression (x³ + y³).

By using the AM-GM inequality, which states that the arithmetic mean of a set of non-negative numbers is always greater than or equal to the geometric mean, we have:

`\[ (x^3 + y^3)/(2) \geq \sqrt[2]{x^3 \cdot y^3} \]`

Simplifying, we get:

`\[ x^3 + y^3 \geq 2\sqrt[2]{(xy)^3} \]`

Substituting xy = 64, we get:

`\[ x^3 + y^3 \geq 2\sqrt[2]{64^3} = 128 \]`

Equality occurs when x = y, which means the minimum value of x³ + y³ is achieved when both numbers are equal to the cube root of 64, which is 4. Therefore, the minimum sum is 4³ + 4³ = 128.

This approach utilizes the concept of AM-GM inequality, a fundamental tool in optimization problems, to establish the minimum value of the sum of cubes under the given constraint.