How do you find the interval where f is concave up and where f is concave down for f(x)?

To determine the intervals where a function f(x) is concave up or concave down, you’ll need to follow these steps:

1. Find the second derivative: Start by calculating the first derivative f'(x) of the function. Then, differentiate f'(x) to obtain the second derivative f”(x).
2. Set the second derivative to zero: Solve the equation f”(x) = 0. This will give you critical points where the concavity of the function may change.
3. Test intervals: Use the critical points to divide the number line into intervals. Choose a test point from each interval and evaluate the second derivative f”(x) at those points.
4. Determine concavity: If f”(x) > 0 in an interval, then the function is concave up on that interval. If f”(x) < 0, the function is concave down on that interval.

For example, if f(x) = x^3 – 3x, the steps would look like this:

1. First derivative: f'(x) = 3x^2 – 3.
2. Second derivative: f”(x) = 6x.
3. Set to zero: 6x = 0 gives x = 0.
4. Test intervals: Choose points from intervals (-∞, 0) and (0, ∞). For x = -1, f”(-1) = -6 (concave down). For x = 1, f”(1) = 6 (concave up).

Thus, f(x) is concave down on (-∞, 0) and concave up on (0, ∞).