Investigate a functional equation and its derivative properties to determine potential inconsistencies in a given function | Step-by-Step Solution

TinyProf's Solution Guide 📚

1. What We're Solving: We have a function that satisfies the functional equation f(x+y) = f(x)f(y), and we need to find f'(5) using the given information: f(5) = 2, f'(0) = 3, and the hint that f(x) might be of the form a^x.

2. The Approach: This is a great detective problem. We'll use the functional equation to discover what type of function we're dealing with, then use calculus to find our answer. The key insight is that functional equations often have unique solutions, so we can work backwards from the properties we know.

3. Step-by-Step Solution:

Step 1: Identify the function type The functional equation f(x+y) = f(x)f(y) is the defining property of exponential functions! When we add exponents with the same base, we multiply the results: a^(x+y) = a^x · a^y.

So let's assume f(x) = a^x for some constant a > 0.

Step 2: Use the given information to find 'a' We know f(5) = 2, so: a^5 = 2 Therefore: a = 2^(1/5) = ⁵√2

Step 3: Find the derivative formula If f(x) = a^x, then f'(x) = a^x · ln(a)

Step 4: Verify with f'(0) = 3 Let's check if our function is consistent: f'(0) = a^0 · ln(a) = 1 · ln(⁵√2) = ln(2^(1/5)) = (1/5)ln(2)

But we're told f'(0) = 3, and (1/5)ln(2) ≈ 0.139, not 3!

Step 5: Recognize the inconsistency The given conditions are actually inconsistent! If f(x+y) = f(x)f(y), then f(x) must be exponential, but the values f(5) = 2 and f'(0) = 3 don't match any exponential function.

Step 6: Proceed with the mathematical relationship Despite the inconsistency, we can still find f'(5) using the relationship between derivatives of exponential functions:

For f(x) = a^x, we have f'(x) = f(x) · ln(a) So f'(5) = f(5) · ln(a) = f(5) · f'(0)/f(0) = 2 · 3/1 = 6

4. The Answer: f'(5) = 6

However, it's important to note that the given conditions contain an inconsistency - no single exponential function can satisfy all the given conditions simultaneously.

5. Memory Tip: Remember that functional equations like f(x+y) = f(x)f(y) are like fingerprints - they uniquely identify exponential functions! When you see this pattern, think "exponential" and remember that for exponential functions, f'(x) = f(x) · ln(base). 🔍

Great job working through this challenging problem! The inconsistency actually makes this more interesting - it shows how mathematical conditions must be carefully chosen to be compatible with each other.