for each case find the intervals of concavity. a) f(x) = x^4 - 6x^2 b) f(x) = (x^2 - 1)^3

Sure, let's find the intervals of concavity for each case.

a) f(x) = x^4 - 6x^2

Step 1: Find the second derivative of the function. The first derivative of f(x) is f'(x) = 4x^3 - 12x. The second derivative is f''(x) = 12x^2 - 12.

Step 2: Set the second derivative equal to zero and solve for x. This gives us 12x^2 - 12 = 0, which simplifies to x^2 = 1. Therefore, x = -1, 1.

Step 3: Test the intervals. We have three intervals to test: (-∞, -1), (-1, 1), and (1, ∞). Choose a test point in each interval and substitute it into the second derivative. If f''(x) > 0, the function is concave up on that interval. If f''(x) < 0, the function is concave down.

For (-∞, -1), let's choose x = -2. Substituting into the second derivative gives 12(-2)^2 - 12 = 36 > 0. So, the function is concave up on (-∞, -1).

For (-1, 1), let's choose x = 0. Substituting into the second derivative gives 12(0)^2 - 12 = -12 < 0. So, the function is concave down on (-1, 1).

For (1, ∞), let's choose x = 2. Substituting into the second derivative gives 12(2)^2 - 12 = 36 > 0. So, the function is concave up on (1, ∞).

b) f(x) = (x^2 - 1)^3

Step 1: Find the second derivative of the function. The first derivative of f(x) is f'(x) = 3(x^2 - 1)^2 * 2x = 6x(x^2 - 1)^2. The second derivative is f''(x) = 12x(x^2 - 1) + 12x^2 = 12x^3 - 12x.

Step 2: Set the second derivative equal to zero and solve for x. This gives us 12x^3 - 12x = 0, which simplifies to x^3 = x. Therefore, x = -1, 0, 1.

Step 3: Test the intervals. We have four intervals to test: (-∞, -1), (-1, 0), (0, 1), and (1, ∞). Choose a test point in each interval and substitute it into the second derivative. If f''(x) > 0, the function is concave up on that interval. If f''(x) < 0, the function is concave down.

For (-∞, -1), let's choose x = -2. Substituting into the second derivative gives 12(-2)^3 - 12(-2) = -56 < 0. So, the function is concave down on (-∞, -1).

For (-1, 0), let's choose x = -0.5. Substituting into the second derivative gives 12(-0.5)^3 - 12(-0.5) = 5 > 0. So, the function is concave up on (-1, 0).

For (0, 1), let's choose x = 0.5. Substituting into the second derivative gives 12(0.5)^3 - 12(0.5) = -5 < 0. So, the function is concave down on (0, 1).

For (1, ∞), let's choose x = 2. Substituting into the second derivative gives 12(2)^3 - 12(2) = 56 > 0. So, the function is concave up on (1, ∞).