To find $f(4)$ given that $f'(x) = \frac{\sqrt{x} - 2}{x^4}$ and $f(1) = 0$, we first need to integrate $f'(x)$ to obtain $f(x)$.

1. Integrate $f'(x)$:

$f'(x) = \frac{\sqrt{x} - 2}{x^4} = \frac{x^{1/2}}{x^4} - \frac{2}{x^4} = x^{-7/2} - 2x^{-4}$

Now, we can integrate each term separately:

$f(x) = \int \left( x^{-7/2} - 2x^{-4} \right) \, dx$

Calculating the integrals:

$\int x^{-7/2} \, dx = \frac{x^{-5/2}}{-5/2} = -\frac{2}{5} x^{-5/2}$

$\int -2x^{-4} \, dx = -2 \cdot \frac{x^{-3}}{-3} = \frac{2}{3} x^{-3}$

So we have:

$f(x) = -\frac{2}{5} x^{-5/2} + \frac{2}{3} x^{-3} + C$

where $C$ is a constant.

2. Find the constant $C$ using the initial condition $f(1) = 0$:

Substituting $x = 1$:

$f(1) = -\frac{2}{5} \cdot 1^{-5/2} + \frac{2}{3} \cdot 1^{-3} + C = 0$

This simplifies to:

$-\frac{2}{5} + \frac{2}{3} + C = 0$

Finding a common denominator (15):

$-\frac{6}{15} + \frac{10}{15} + C = 0$

This gives:

$C = -\frac{4}{15}$

So the function $f(x)$ is:

$f(x) = -\frac{2}{5} x^{-5/2} + \frac{2}{3} x^{-3} - \frac{4}{15}$

3. Now, evaluate $f(4)$:

Substituting $x = 4$:

$f(4) = -\frac{2}{5} (4^{-5/2}) + \frac{2}{3} (4^{-3}) - \frac{4}{15}$

Calculating each term:

$4^{-5/2} = \frac{1}{(4^{5/2})} = \frac{1}{32}$

$4^{-3} = \frac{1}{64}$

Thus,

$f(4) = -\frac{2}{5} \cdot \frac{1}{32} + \frac{2}{3} \cdot \frac{1}{64} - \frac{4}{15}$

Calculating the first term:

$-\frac{2}{5 \cdot 32} = -\frac{2}{160} = -\frac{1}{80}$

Calculating the second term:

$\frac{2}{3 \cdot 64} = \frac{2}{192} = \frac{1}{96}$

Now we have:

$f(4) = -\frac{1}{80} + \frac{1}{96} - \frac{4}{15}$

Finding a common denominator for $80$, $96$, and $15$, which is $480$:

Converting each fraction:

$-\frac{1}{80} = -\frac{6}{480}, \quad \frac{1}{96} = \frac{5}{480}, \quad -\frac{4}{15} = -\frac{128}{480}$

So:

$f(4) = -\frac{6}{480} + \frac{5}{480} - \frac{128}{480} = \frac{-6 + 5 - 128}{480} = \frac{-129}{480}$

Thus, the final result is:

$f(4) = -\frac{129}{480}$

Would you like further details or have any questions? Here are some related questions you might consider:

1. How do you find $f'(x)$ from $f(x)$?
2. What are the implications of the constant $C$ in the integration process?
3. Can you explain the process of finding a common denominator in detail?
4. How would you approach finding $f(a)$ for other values of $a$?
5. What is the significance of the initial condition in this problem?

Tip: Always check your integration constants against given conditions to ensure correctness.