Prove that
a^4 + b^4 + c^4 ¡Ý b^2ac + c^2ab
holds for all real numbers a, b and c.
To prove the inequality a^4 + b^4 + c^4 ≥ b^2ac + c^2ab for all real numbers a, b, and c, we can use the AM-GM inequality.
The AM-GM inequality states that for non-negative real numbers x and y, the following inequality holds:
2√(x * y) ≤ (x + y)
Applying this inequality to the terms b^4 and c^4, we have:
2√(b^4 * c^4) ≤ (b^4 + c^4)
Simplifying this further:
2b^2c^2 ≤ (b^4 + c^4)
Similarly, applying the AM-GM inequality to the terms a^2b and a^2c, we have:
2√(a^2b * a^2c) ≤ (a^2b + a^2c)
Simplifying this further:
2a^2bc ≤ (a^4 + b^4)
Now, adding the two inequalities together, we have:
2b^2c^2 + 2a^2bc ≤ (b^4 + c^4) + (a^4 + b^4)
Simplifying:
2b^2c^2 + 2a^2bc ≤ a^4 + b^4 + c^4 + b^4
Rearranging the terms:
a^4 + b^4 + c^4 ≥ b^2ac + c^2ab
Thus, we have proved that the inequality holds for all real numbers a, b, and c.