To prove that $f(x+y) \leq f(x) + f(y)$ for a function $f$ defined on $[0, \infty)$ such that $f: [0, \infty) \rightarrow [0, \infty)$ and $f''(x) \leq 0$ for all $x \in [0, \infty)$, we can use the fact that $f''(x) \leq 0$ implies that the function is concave.

Proof:

1. Concavity from $f''(x) \leq 0$: Since $f''(x) \leq 0$ for all $x$, $f$ is a concave function. This means that the secant line between any two points on the graph of $f$ lies above the graph. Formally, for any $x, y \geq 0$ and $t \in [0, 1]$, we have: $f(tx + (1-t)y) \geq tf(x) + (1-t)f(y).$

2. Setting $t = \frac{1}{2}$: Choose $t = \frac{1}{2}$. Then for any $x, y \geq 0$: $f\left(\frac{x+y}{2}\right) \geq \frac{1}{2}f(x) + \frac{1}{2}f(y).$ Multiply both sides by 2: $2f\left(\frac{x+y}{2}\right) \geq f(x) + f(y).$

3. Jensen's Inequality: For concave functions, Jensen's inequality implies that for any $x_1, x_2, \ldots, x_n \in [0, \infty)$ and non-negative weights $\lambda_1, \lambda_2, \ldots, \lambda_n$ such that $\sum_{i=1}^n \lambda_i = 1$: $f\left(\sum_{i=1}^n \lambda_i x_i\right) \geq \sum_{i=1}^n \lambda_i f(x_i).$ In our case, we are considering only two points $x$ and $y$ with equal weights $\lambda_1 = \lambda_2 = \frac{1}{2}$.

4. Conclusion: The inequality $2f\left(\frac{x+y}{2}\right) \geq f(x) + f(y)$ suggests that the function $f(x)$ is subadditive, meaning: $f(x + y) \leq f(x) + f(y).$ This completes the proof that for a concave function defined on $[0, \infty)$ with $f''(x) \leq 0$, the inequality $f(x+y) \leq f(x) + f(y)$ holds.

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Here are 8 related questions that might interest you:

1. What are the implications of concavity for other functional inequalities?
2. How does Jensen's inequality relate to convex and concave functions?
3. Can you provide examples of functions that satisfy $f(x+y) \leq f(x) + f(y)$?
4. How does the condition $f''(x) \leq 0$ affect the graph of the function $f(x)$?
5. What are the applications of subadditive functions in economics or optimization?
6. How would the proof change if $f(x)$ were convex instead of concave?
7. What is the relationship between concavity and the second derivative test?
8. How can we generalize this proof to higher dimensions?

Tip: When working with inequalities involving derivatives, always consider the implications of the function's concavity or convexity for establishing bounds and inequalities.