f is a function defined on [0, inf) -> [0,inf) and f'' <= 0. for all x in [0, inf). prove that f(x+y) <= f(x) +f(y)

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.

Learn how to prove f(x+y) <= f(x) + f(y) for a function f defined on [0, ∞) such that f''(x) ≤ 0 for all x in [0, ∞). This proof involves understanding concave functions and applying Jensen's inequality. Suitable for advanced undergraduate students and those interested in mathematical analysis.

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