To determine the intervals on which $f(x) = \ln(6x^2 + 1)$ is concave up or concave down, and to locate any inflection points, we’ll need to calculate the second derivative of $f(x)$.

Step 1: Find the First Derivative

The first derivative of $f(x) = \ln(6x^2 + 1)$ is: $f'(x) = \frac{d}{dx} \ln(6x^2 + 1) = \frac{1}{6x^2 + 1} \cdot 12x = \frac{12x}{6x^2 + 1}$

Step 2: Find the Second Derivative

Now, we’ll differentiate $f'(x) = \frac{12x}{6x^2 + 1}$ to get $f''(x)$ and analyze concavity: Using the quotient rule, we have: $f''(x) = \frac{(6x^2 + 1) \cdot 12 - 12x \cdot 12x}{(6x^2 + 1)^2}$ Simplifying the numerator: $f''(x) = \frac{12(6x^2 + 1) - 144x^2}{(6x^2 + 1)^2} = \frac{72 - 132x^2}{(6x^2 + 1)^2} = \frac{-72x^2 + 12}{(6x^2 + 1)^2}$

Step 3: Set $f''(x) = 0$ to Find Inflection Points

To find the inflection points, we solve $f''(x) = 0$: $-72x^2 + 12 = 0$ $x^2 = \frac{1}{6}$ $x = \pm \frac{1}{\sqrt{6}}$

Thus, there are potential inflection points at $x = \pm \frac{1}{\sqrt{6}}$.

Step 4: Determine Concavity Intervals

To determine where $f(x)$ is concave up or concave down, we check the sign of $f''(x)$ on the intervals divided by the points $x = \pm \frac{1}{\sqrt{6}}$.

1. For $x < -\frac{1}{\sqrt{6}}$: $f''(x) > 0$, so $f(x)$ is concave up.
2. For $-\frac{1}{\sqrt{6}} < x < \frac{1}{\sqrt{6}}$: $f''(x) < 0$, so $f(x)$ is concave down.
3. For $x > \frac{1}{\sqrt{6}}$: $f''(x) > 0$, so $f(x)$ is concave up.

Final Answer:

Part 1

• The function is concave up on $\left( -\infty, -\frac{1}{\sqrt{6}} \right) \cup \left( \frac{1}{\sqrt{6}}, \infty \right)$
• The function is concave down on $\left( -\frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}} \right)$

Part 2

• Inflection points occur at $x = -\frac{1}{\sqrt{6}}$ and $x = \frac{1}{\sqrt{6}}$.

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Would you like further clarification on any steps?

Related Questions:

1. How do you interpret the concavity of a function in terms of its graph?
2. What is the significance of inflection points on a function's curve?
3. How does the second derivative test determine concavity?
4. Can a function have concave up and concave down sections without any inflection points?
5. How does the logarithmic function’s concavity behavior differ from polynomial functions?

Tip:

When finding concavity, remember that intervals of concavity change at points where the second derivative equals zero or is undefined.