Final answer:
To find the intervals where f(x) is increasing or decreasing, take the first derivative and analyze its sign. To find the intervals where f(x) is concave up or concave down, take the second derivative and analyze its sign.
Step-by-step explanation:
To find the intervals where f(x) is increasing or decreasing, we need to find the first derivative of f(x) and analyze its sign. The first derivative of f(x) = e^(-x^2) is f'(x) = -2xe^(-x^2). Now, we can analyze the sign of f'(x) to determine the intervals where f(x) is increasing or decreasing. The interval where f(x) is increasing is when f'(x) > 0, which occurs when x < 0. The interval where f(x) is decreasing is when f'(x) < 0, which occurs when x > 0.
To find the intervals where f(x) is concave up or concave down, we need to find the second derivative of f(x) and analyze its sign. The second derivative of f(x) is f''(x) = 2(2x^2 - 1)e^(-x^2). Now, we can analyze the sign of f''(x) to determine the intervals where f(x) is concave up or concave down. The interval where f(x) is concave up is when f''(x) > 0, which occurs when -1 < x < 1. The interval where f(x) is concave down is when f''(x) < 0, which occurs when x < -1 or x > 1.