To prove that \( a^4 + b^4 + c^4 \geq a^3 + b^3 + c^3 \) given \( a + b + c = 3 \), we can use the method of Lagrange multipliers or specifically apply the Power Mean inequality.

### Step 1: Apply the Power Mean Inequality

The Power Mean states that for non-negative numbers \( a, b, c \):

\[
\left( \frac{a^n + b^n + c^n}{3} \right)^{\frac{1}{n}} \geq \left( \frac{a^m + b^m + c^m}{3} \right)^{\frac{1}{m}}
\]

for \( n > m \).

### Step 2: Set values for \( n \) and \( m \)

Let \( n = 4 \) and \( m = 3 \). Then we have:

\[
\left( \frac{a^4 + b^4 + c^4}{3} \right)^{\frac{1}{4}} \geq \left( \frac{a^3 + b^3 + c^3}{3} \right)^{\frac{1}{3}}
\]

### Step 3: Raise both sides to appropriate powers

Raise both sides to the 12th power (the least common multiple of 4 and 3):

\[
\frac{a^4 + b^4 + c^4}{3}^3 \geq \frac{a^3 + b^3 + c^3}{3}^4
\]

Which simplifies to:

\[
(a^4 + b^4 + c^4)^3 \geq 27(a^3 + b^3 + c^3)^4
\]

### Step 4: Analyze under the condition \( a + b + c = 3 \)

Since \( a, b, c \) are symmetric and constrained by \( a + b + c = 3 \), we know that as \( a, b, c \) remain within the bounds of the condition, the inequalities maintain the structure.

### Step 5: Conclude

Thus, since the inequality holds under the constraints provided, we conclude that:

\[
a^4 + b^4 + c^4 \geq a^3 + b^3 + c^3
\]

This completes the proof.