Question

If \(a, b, c\) and \(d\) are four positive numbers such that \(a+b+c+d=4\), then what is the maximum value of \((a+1)(b+1)(c+1)(d+1) ?\) (a) 32 (b) 8 (c) 16 (d) 81

Short answer

Answer: 16

Step-by-step solution

Apply the AM-GM Inequality

Because a, b, c, and d are positive numbers, we know that a+1, b+1, c+1, and d+1 are also positive numbers. We will apply the AM-GM inequality on a+1, b+1, c+1, and d+1: $$\frac{(a+1)+(b+1)+(c+1)+(d+1)}{4}\geq \sqrt[4]{(a+1)(b+1)(c+1)(d+1)}$$

Simplify the expression

Now, let's simplify the left side of the inequality: $$\frac{a+b+c+d+4}{4}\geq \sqrt[4]{(a+1)(b+1)(c+1)(d+1)}$$ Since we know that \(a+b+c+d=4\), we substitute it in the left side: $$\frac{4+4}{4}\geq \sqrt[4]{(a+1)(b+1)(c+1)(d+1)}$$ Simplify to get: $$\frac{8}{4}\geq \sqrt[4]{(a+1)(b+1)(c+1)(d+1)}$$ So, \(2 \geq \sqrt[4]{(a+1)(b+1)(c+1)(d+1)}\)

Take the fourth power of both sides

To get rid of the fourth root, we take the fourth power of both sides: $$(2)^4\geq (a+1)(b+1)(c+1)(d+1)$$ $$16 \geq (a+1)(b+1)(c+1)(d+1)$$ So, the maximum value of \((a+1)(b+1)(c+1)(d+1)\) is 16. Therefore, the answer is (c) 16.

Key concepts

Inequality Theorems

The AM-GM Inequality is one of the most important inequality theorems in mathematics. It establishes a relationship between the arithmetic mean (AM) and the geometric mean (GM) of a set of non-negative numbers. This theorem states that the arithmetic mean of a list of non-negative numbers is always greater than or equal to the geometric mean of the same list. In mathematical terms, if you have n non-negative numbers, say \(x_1, x_2, \ldots, x_n\), the AM-GM Inequality is given by:

• \(\frac{x_1 + x_2 + \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdot x_2 \cdot \ldots \cdot x_n}\)

This relationship becomes an equality (i.e., both sides are equal) when all the numbers \(x_1, x_2, \ldots, x_n\) are the same. It's particularly useful in finding the maximum or minimum values in various optimization problems, as seen in the exercise with variables •a, b, c, d• constrained by their sum to maximize another expression.

Arithmetic Mean

The arithmetic mean (AM) is a measure of central tendency. It is simply the sum of a set of values divided by the number of values. In essence, it provides an average, or middle point, of the numbers you are working with. The formula is written as follows:

• \( \text{AM} = \frac{x_1 + x_2 + \ldots + x_n}{n} \)

This is fundamental when applying the AM-GM Inequality, as it represents the average value of the numbers you’re comparing. In the context of the exercise, this understanding helps us comprehend why setting the arithmetic mean of \((a+1), (b+1), (c+1), (d+1)\) will remain fixed under transformation, allowing us to bound the expression by its maximum potential. This concept helps set the stage for using inequality theorems to verify solution outcomes.

Geometric Mean

The geometric mean (GM) is another type of average that is especially useful in situations involving growth rates and multiplicative processes. It is calculated by taking the n-th root of the product of n values. Here's the formula:

• \( \text{GM} = \sqrt[n]{x_1 \cdot x_2 \cdot \ldots \cdot x_n} \)

In problem-solving contexts, the GM is relevant when you're dealing with ratios or products, such as in scenarios where constant percentage changes are applied. In the given problem, the GM of the numbers \((a+1), (b+1), (c+1), (d+1)\) is calculated and compared to the AM using the AM-GM inequality to deduce that their product must have an upper bound. The geometric mean embodies an equal distribution when the product potential is maximized under given constraints.

Problem-Solving Techniques

When approaching optimization problems, especially those involving inequalities, problem-solving techniques like applying known theorems are crucial. With the problem at hand, the use of the AM-GM Inequality is pivotal. Here's a general approach:

• Identify constraints: Understand the limitations set by the problem. Here, it's \(a + b + c + d = 4\).
• Apply relevant theorems: Use the AM-GM Inequality to find bounds or maximum/minimum values.
• Simplify expressions: Manipulate the given equations to facilitate easier comparison or calculation.
• Verification: Once the theoretical max or min is found, verify through substitution or logical reasoning.

These techniques simplify complex problems systematically, making them more approachable even if certain aspects initially seem challenging. By mastering these strategies, tackling similar exercises becomes intuitive, as demonstrated by simplifying and bounding the function in the tutorial solution.