To determine the concavity of the function $f(x) = \sqrt{4x} \ln(x)$, we need to calculate the second derivative and analyze its sign.

Step 1: Find the first derivative $f'(x)$

Let's rewrite $f(x)$ to make differentiation simpler: $f(x) = \sqrt{4x} \ln(x) = 2\sqrt{x} \ln(x) = 2x^{1/2} \ln(x)$

Using the product rule for differentiation, where $u = 2x^{1/2}$ and $v = \ln(x)$:

1. $u' = \frac{1}{\sqrt{x}}$
2. $v' = \frac{1}{x}$

Then, $f'(x) = u'v + uv' = \left( \frac{1}{\sqrt{x}} \right) \ln(x) + 2x^{1/2} \cdot \frac{1}{x} = \frac{\ln(x)}{\sqrt{x}} + \frac{1}{\sqrt{x}} = \frac{\ln(x) + 1}{\sqrt{x}}$

Step 2: Find the second derivative $f''(x)$

Now, differentiate $f'(x) = \frac{\ln(x) + 1}{\sqrt{x}}$ again.

Using the quotient rule for differentiation, where:

• Numerator: $g(x) = \ln(x) + 1$, so $g'(x) = \frac{1}{x}$
• Denominator: $h(x) = \sqrt{x} = x^{1/2}$, so $h'(x) = \frac{1}{2\sqrt{x}}$

The second derivative $f''(x)$ is given by: $f''(x) = \frac{g' h - g h'}{h^2} = \frac{\left(\frac{1}{x}\right) \sqrt{x} - (\ln(x) + 1) \cdot \frac{1}{2\sqrt{x}}}{x}$ Simplifying each term will show where $f''(x)$ is positive (concave up) or negative (concave down).

Step 3: Determine concavity and inflection points

1. Solve $f''(x) = 0$ to find possible inflection points.
2. Analyze the intervals based on where $f''(x)$ changes sign.

Once calculated, we can determine whether to select:

• Option A if the function is both concave up and concave down on different intervals.
• Option B or C if the function is only concave up or concave down across all intervals.
• Option D if the function has no concavity.

Would you like further assistance with detailed interval calculations or the specific selection of intervals for concavity? Here are related questions you might find helpful:

1. What is the process of finding intervals of concavity for any given function?
2. How does the presence of a logarithmic term affect concavity?
3. How do inflection points relate to the behavior of the first derivative?
4. What are typical points where functions with square roots or logarithms change concavity?
5. How does interval notation help in expressing concavity?

Tip: When working with functions involving $\ln(x)$, ensure the domain $x > 0$ is respected for valid solutions.