Question:

Step 1: Understand the given constraint on the function. We are given that $|f(x) - f(y)| \leq 6|x - y|^2$ for all $x, y \in \mathbb{R}$. This is a Lipschitz-type condition that bounds the difference in function values by a quadratic term in the distance between inputs. Step 2: Divide both sides by the distance between points. For $x \neq y$, we can divide both sides of the inequality by $|x - y|$: $$\frac{|f(x) - f(y)|}{|x - y|} \leq 6|x - y|$$ This expresses the average rate of change of $f$ between two points in terms of their distance. Step 3: Take the limit as the two points approach each other. Taking the limit as $y \to x$ on both sides: $$\lim_{y \to x} \frac{|f(x) - f(y)|}{|x - y|} \leq \lim_{y \to x} 6|x - y|$$ The left side is the definition of $|f'(x)|$ (the absolute value of the derivative), and the right side approaches $6 \cdot 0 = 0$: $$|f'(x)| \leq 0$$ Step 4: Conclude that the derivative is zero everywhere. Since the absolute value of the derivative is non-negative and bounded above by zero, we must have: $$f'(x) = 0 \text{ for all } x \in \mathbb{R}$$ This means the function has zero slope at every point. Step 5: Determine that f is a constant function. If $f'(x) = 0$ for all $x$, then $f$ is a constant function. That is, $f(x) = c$ for some constant $c$ and all $x \in \mathbb{R}$. Step 6: Use the given condition to find the constant. We are given that $f(3) = 6$. Since $f$ is constant: $$f(x) = 6 \text{ for all } x \in \mathbb{R}$$ Step 7: Calculate f(6). Therefore: $$f(6) = 6$$ The answer is **Option 4: 6**.