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To solve the problem, let’s analyze and integrate the given expression:

We are asked to find the most general antiderivative of:

\[
\int \left(\frac{\sqrt{y}}{4} - \frac{4}{\sqrt{y}}\right) \, dy
\]

Let's break down the integral by separating it into two parts:

\[
\int \left(\frac{\sqrt{y}}{4} - \frac{4}{\sqrt{y}}\right) \, dy = \int \frac{\sqrt{y}}{4} \, dy - \int \frac{4}{\sqrt{y}} \, dy
\]

### Step 1: Solve \(\int \frac{\sqrt{y}}{4} \, dy\)

Rewrite \(\sqrt{y}\) as \(y^{1/2}\):

\[
\int \frac{y^{1/2}}{4} \, dy = \frac{1}{4} \int y^{1/2} \, dy
\]

Using the power rule for integration, \(\int y^n \, dy = \frac{y^{n+1}}{n+1}\) for \(n \neq -1\), we get:

\[
= \frac{1}{4} \cdot \frac{y^{3/2}}{3/2} = \frac{1}{4} \cdot \frac{2}{3} y^{3/2} = \frac{1}{6} y^{3/2}
\]

### Step 2: Solve \(\int \frac{4}{\sqrt{y}} \, dy\)

Rewrite \(\frac{4}{\sqrt{y}}\) as \(4 y^{-1/2}\):

\[
\int 4 y^{-1/2} \, dy = 4 \int y^{-1/2} \, dy
\]

Using the power rule again, we get:

\[
= 4 \cdot \frac{y^{1/2}}{1/2} = 4 \cdot 2 y^{1/2} = 8 y^{1/2}
\]

### Step 3: Combine the results

Now we can put it all together:

\[
\int \left(\frac{\sqrt{y}}{4} - \frac{4}{\sqrt{y}}\right) \, dy = \frac{1}{6} y^{3/2} - 8 y^{1/2} + C
\]

### Conclusion

The answer matches option **B**:

\[
\frac{1}{6} y^{3/2} - 8 y^{1/2} + C
\]

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Would you like further details on the steps? Here are some related questions for more practice:

1. How do you apply the power rule for integration?
2. What happens when integrating functions with fractional exponents?
3. How do you verify antiderivatives by differentiation?
4. How do you deal with constants when integrating?
5. What are common techniques for solving integrals involving radicals?

**Tip:** When integrating functions with radicals, try rewriting the radical in exponential form to make applying the power rule easier.

Find the most general antiderivative of \( \int \left(\frac{\sqrt{y}}{4} - \frac{4}{\sqrt{y}}\right) \, dy \).

Explore the step-by-step solution for finding the most general antiderivative of the integral \( \int \left(\frac{\sqrt{y}}{4} - \frac{4}{\sqrt{y}}\right) \, dy \). This problem demonstrates integration techniques involving radicals and applying the power rule. Suitable for Grades 11-12.

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