f”(x) is negative
f(x) is concave down & f'(x) is decreasing
If the second derivative f”(x) is negative for a function f(x), it means that the rate of change of the slope of the function is decreasing. In other words, the function is concave down or is curving downwards at the given point.
This can be visualized as a bowl shape or a parabola that opens downwards. At the point where the second derivative is negative, the function is at its maximum.
For example, consider the function f(x) = -x^2. The first derivative f'(x) = -2x represents the slope of the function at each point. The second derivative of the function is f”(x) = -2, which is a negative constant.
This means that the slope of the function is decreasing at all points, and the function is concave down. The function has a maximum at x = 0. This can be verified by setting the first derivative equal to zero, solving for x, and plugging the value of x back into the original function.
Overall, knowing that the second derivative f”(x) is negative helps in identifying important points of a function such as maxima and concavity.
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