f”(x) is negative
f(x) is concave down & f'(x) is decreasing
When we say that f”(x) is negative, we are indicating that the second derivative of the function f(x) is negative at point x. In other words, the rate of change of the slope of the function f(x) is decreasing at point x. This gives us important information about the concavity of the function f(x).
If the second derivative f”(x) is negative, then the function f(x) is said to be concave down at point x. This means that the graph of the function will be curving downwards, like the shape of a frown. In particular, any local maximum point of the function will occur at the point where the second derivative changes sign from negative to positive.
Furthermore, if we know that the second derivative is negative for all points x in a certain interval, then we can conclude that the function f(x) is concave down over that entire interval. This information can be useful for analyzing the behavior of the function, including finding the locations of any inflection points or anticipating the general shape of the graph.
More Answers:
Inflection Points In Calculus: Analysis Of Function BehaviorThe Relationship Between Second Derivative And Critical Points In Calculus.
Concavity And The Second Derivative For Function F(X)