If f(x) is concave down, then f”(x) is?
If a function f(x) is concave down, it means that its graph is shaped like a downward-facing U or a bowl, curving downwards
If a function f(x) is concave down, it means that its graph is shaped like a downward-facing U or a bowl, curving downwards. This concavity gives us information about the second derivative of the function, f”(x).
The second derivative, f”(x), represents the rate at which the slope of the function is changing. If f(x) is concave down, it means that its slope is decreasing as x increases. In other words, the rate at which the function is getting steeper is decreasing.
Mathematically, if f”(x) > 0, it means that the function is concave up, with its graph shaped like an upward-facing U or a bowl, curving upwards. On the other hand, if f”(x) < 0, it means that the function is concave down, with its graph shaped like a downward-facing U or a bowl, curving downwards. Therefore, if f(x) is concave down, f''(x) must be negative (f''(x) < 0) to reflect the decreasing slope and concave downward shape of the function.
More Answers:
The Relationship Between Increasing Functions and Positive Derivatives in MathematicsUnderstanding the Relationship Between Decreasing Functions and Negative Derivatives
Understanding Concavity: Exploring the Relationship between the Second Derivative and Graph Shape