To prove that $f(8) > 2024 \cdot f(0)$ given that $f$ is a differentiable function on $\mathbb{R}$ with $f'(x) > f(x) > 0$ for all $x$, let's proceed step by step.

Step 1: Establishing that $f$ is an increasing function

Since $f'(x) > f(x) > 0$, this implies that:

• $f(x) > 0$ for all $x$, so $f(x)$ is always positive.
• $f'(x) > 0$, meaning that $f(x)$ is strictly increasing on $\mathbb{R}$.

Step 2: Relating $f'(x)$ and $f(x)$

The fact that $f'(x) > f(x)$ for all $x$ implies that the rate of growth of $f(x)$ exceeds its current value. We can use this inequality to deduce something about the relative sizes of $f(x)$ at different points.

Step 3: Exponential-like behavior of $f(x)$

Since $f'(x) > f(x)$, the function grows faster than an exponential function. Consider the differential inequality:

$f'(x) > f(x)$

This inequality suggests that $f(x)$ grows at least as fast as an exponential function. Specifically, if we had $f'(x) = f(x)$, the solution to this equation would be $f(x) = Ce^x$ for some constant $C$.

However, since $f'(x) > f(x)$, the function $f(x)$ grows even faster than an exponential function of the form $Ce^x$.

Step 4: Bounding the growth of $f(x)$

To make this more precise, let's compare $f(x)$ to the solution of the differential equation $f'(x) = f(x)$. The solution to this equation, as we mentioned, is $f(x) = f(0)e^x$.

Thus, we have:

$f(x) > f(0) e^x$

for all $x$, because $f'(x) > f(x)$ implies that $f(x)$ grows faster than $f(0)e^x$.

Step 5: Applying this bound to the given problem

We need to show that $f(8) > 2024 \cdot f(0)$. Using the inequality $f(x) > f(0) e^x$, we evaluate this at $x = 8$:

$f(8) > f(0) e^8$

Now, we need to check whether $e^8 > 2024$. Using an approximation, we know that:

$e^8 \approx 2980.96$

Since $2980.96 > 2024$, we conclude that:

$f(8) > f(0) \cdot 2980.96 > 2024 \cdot f(0)$

Conclusion

We have shown that $f(8) > 2024 \cdot f(0)$, completing the proof.

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Would you like any further details on this proof, or do you have any questions?

Here are five related questions to deepen your understanding:

1. How would the result change if we only knew that $f'(x) \geq f(x)$?
2. Can we determine the exact form of $f(x)$ from the given conditions?
3. What happens if $f'(x) < f(x)$? How would that affect the function's growth?
4. How can we apply similar methods to solve other inequalities involving derivatives?
5. What does this result tell us about the long-term behavior of $f(x)$ as $x \to \infty$?

Tip: Understanding how differential inequalities relate to the growth of functions can be a powerful tool when dealing with problems that involve unknown functions and their rates of change.