To solve this problem logically, we need to analyze the inequality given:

$(a + b)(a - b)^2 < 0 \iff (a + b)^3 > 4(a^3 + b^3)$

Let's go through each side to understand the conditions needed for this inequality to hold.

Step 1: Analyzing $(a + b)(a - b)^2 < 0$

1. The term $(a - b)^2$ is a square, so it is always non-negative. Therefore, $(a - b)^2 \geq 0$.

2. For the product $(a + b)(a - b)^2$ to be negative (as the inequality suggests, $(a + b)(a - b)^2 < 0$), we require:

   • $(a + b)$ to be negative.
   • $(a - b)^2$ to be positive (so $a \neq b$, otherwise it would be zero).

   Hence, the condition $(a + b) < 0$ and $a \neq b$ must hold for the left side of the inequality to be true.

Step 2: Analyzing $(a + b)^3 > 4(a^3 + b^3)$

To proceed, let’s expand $(a + b)^3$ and analyze it against $4(a^3 + b^3)$.

1. Expanding $(a + b)^3$: $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

2. Setting up the inequality: We now need: $a^3 + 3a^2b + 3ab^2 + b^3 > 4a^3 + 4b^3$ Simplifying, this becomes: $3a^2b + 3ab^2 > 3a^3 + 3b^3$ Dividing through by 3: $a^2b + ab^2 > a^3 + b^3$

3. Checking if this condition holds: This inequality may hold for certain values of $a$ and $b$ where $a + b$ is negative, and specifically, it’s more likely to hold if both $a$ and $b$ are negative but not equal.

Conclusion

Thus, the logical requirement for this inequality to hold is:

• $a + b < 0$ (which implies both are negative or one is negative and the other is of smaller magnitude in positive).
• $a \neq b$.

This logical deduction shows that if these conditions are met, then the inequality holds as desired.

Would you like further details, or do you have any questions?

Related Questions

1. How does the inequality change if $a$ and $b$ are both positive?
2. What happens if $a$ and $b$ are equal?
3. Can this inequality hold if $a$ is positive and $b$ is negative?
4. How can we use other techniques (e.g., calculus) to verify this inequality?
5. What are examples of specific values of $a$ and $b$ that satisfy this inequality?

Tip

When tackling inequalities involving powers and products, analyzing sign conditions of each factor often simplifies the process of finding when the inequality holds.