To find the maximum value of the expression \( ab + bc + cd + da \) given the constraint \( a + b + c + d = 1 \), we can utilize the method of Lagrange multipliers or apply inequalities such as the Cauchy-Schwarz inequality. Here’s a detailed look into the problem.
Understanding the Expression
The expression \( ab + bc + cd + da \) can be rearranged as:
- \( ab + da + bc + cd \)
This represents the sum of products of adjacent variables.
Applying the AM-GM Inequality
To maximize this expression, we can leverage the Arithmetic Mean-Geometric Mean (AM-GM) inequality, which states that for any non-negative numbers, the arithmetic mean is greater than or equal to the geometric mean.
- Group the terms:
- \( ab + da \)
- \( bc + cd \)
Applying AM-GM gives:
- \( ab + da \leq \frac{(a+d) + (b+b)}{2} \)
- \( bc + cd \leq \frac{(b+c) + (c+b)}{2} \)
Setting Equal Variables
To find the maximum, consider the case where two pairs of variables are equal, for example:
- Let \( a = b \) and \( c = d \).
Then we have:
- \( 2a + 2c = 1 \) leading to \( a + c = \frac{1}{2} \).
Assuming \( a = c = x \):
- We can now rewrite the expression \( 2x^2 + 2x^2 = 4x^2 \).
To maximize \( 4x^2 \) under the constraint \( 2x = \frac{1}{2} \), we find \( x = \frac{1}{4} \).
Final Calculation
Substituting \( x = \frac{1}{4} \):
- \( ab + bc + cd + da = 4 \left(\frac{1}{4}\right)^2 = 4 \times \frac{1}{16} = \frac{1}{4} \).
Thus, the maximum value of \( ab + bc + cd + da \) is:
Maximum Value: 0.25