Step 1: Integral
Given the derivative of the function, F'(x) = 7^x, the first step is to find the original function, f(x). We do this by computing the integral of F'(x), which is 7^x. The integral of this function gives us f(x) = 7^x.
Step 2: Initial Condition
We are given that f(4) = 0. This translates to the condition where if x = 4, the value of the function, f(x) = 0. Therefore we create an equation, 7^4 + C = 0, where C is the constant of integration.
Step 3: Solve for C
Solving the equation obtained in Step 2 for C gives us C = -7^4, which simplifies to C = -2401.
Step 4: Solution
Now, we substitute C into the original function obtained in Step 1. Therefore the final solution is f(x) = 7^x - 2401.
Thus, the function f(x) that is described by the given initial value problem is f(x) = 7^x - 2401.