What are some examples of functions on which the first derivative is concave downwards and the second derivative is also concave downwards?

 A function on which the first derivative is concave downwards and the second derivative is also concave downwards is a function that exhibits a consistent concave downward curvature throughout its entire domain. Here are a few examples:

1. The function f(x) = -x^2: This is a simple quadratic function with a negative leading coefficient. Its first derivative, f'(x) = -2x, is linear and slopes downward. The second derivative, f''(x) = -2, is a constant negative value, indicating a concave downward curvature.

2. The function f(x) = -e^(-x): This is an exponential function with a negative sign, causing it to curve downward. Its first derivative, f'(x) = e^(-x), is a positive exponential function that slopes downward. The second derivative, f''(x) = -e^(-x), is again negative, indicating a consistent concave downward curvature.

3. The function f(x) = -sin(x): This is a trigonometric function with a negative sign, resulting in a downward curvature. Its first derivative, f'(x) = -cos(x), is a cosine function that slopes downward and oscillates between -1 and 1. The second derivative, f''(x) = sin(x), is a sine function that continues to exhibit a downward concave curvature.

These examples demonstrate functions where both the first and second derivatives maintain a consistent concave downward shape throughout their respective domains.