To determine the intervals of concavity and the points of inflection for $f(x) = 2 + \frac{11}{12}x - 2x^2 + \frac{1}{12}x^4$, we will analyze the function's second derivative, $f''(x)$.

Steps:

1. Find $f'(x)$ (the first derivative).
2. Find $f''(x)$ (the second derivative).
3. Solve $f''(x) = 0$ to find possible points of inflection.
4. Determine the concavity by analyzing the sign of $f''(x)$ around these points.

I'll go through these steps now.

1. First Derivative $f'(x)$:

Given: $f(x) = 2 + \frac{11}{12}x - 2x^2 + \frac{1}{12}x^4$

The derivative $f'(x)$ is: $f'(x) = \frac{11}{12} - 4x + \frac{1}{3}x^3$

2. Second Derivative $f''(x)$:

The second derivative $f''(x)$ is: $f''(x) = -4 + x^2$

3. Solve $f''(x) = 0$:

Set $f''(x) = -4 + x^2 = 0$ to find the points where concavity might change: $x^2 = 4$ $x = \pm 2$

So, $x = -2$ and $x = 2$ are potential points of inflection.

4. Determine the Concavity:

To determine intervals of concavity, we check the sign of $f''(x)$ on intervals around $x = -2$ and $x = 2$.

• For $x < -2$, choose a point like $x = -3$: $f''(-3) = -4 + (-3)^2 = -4 + 9 = 5 > 0$ So, $f(x)$ is concave up on $(-\infty, -2)$.

• For $-2 < x < 2$, choose a point like $x = 0$: $f''(0) = -4 + 0^2 = -4 < 0$ So, $f(x)$ is concave down on $(-2, 2)$.

• For $x > 2$, choose a point like $x = 3$: $f''(3) = -4 + 3^2 = -4 + 9 = 5 > 0$ So, $f(x)$ is concave up on $(2, \infty)$.

Summary

• The function is concave up on $(-\infty, -2)$ and $(2, \infty)$.
• The function is concave down on $(-2, 2)$.
• It changes concavity at $x = -2$ and $x = 2$.

Answer

• Concave up on: $(- \infty, -2)$, $(2, \infty)$
• Concave down on: $(-2, 2)$
• Changes concavity at: $x = -2$, $x = 2$

Would you like further details, or do you have any questions?

Here are 5 related questions for further exploration:

1. How do you interpret concavity in terms of a function's shape?
2. What is the significance of points where $f''(x) = 0$?
3. How would you find concavity if $f(x)$ had a different polynomial degree?
4. Can there be cases where $f''(x) = 0$ but the function doesn’t change concavity?
5. What is the relationship between the second derivative and inflection points?

Tip: For concavity questions, always check the sign of the second derivative around critical points where it equals zero.