the graph of f”(x) is below the x-axis at x=2
f(x) is concave down when x=2
The given information tells us that the second derivative of the function f(x) is negative or decreasing at x=2. This means that the curvature of f(x) is concave downwards at x=2.
More specifically, if we take the first derivative of f(x), denoted as f'(x), we can determine whether the function is increasing or decreasing at x=2. This is because the sign of f'(x) will be positive if the function is increasing and negative if the function is decreasing.
To see why this is true, consider the definition of the derivative as the limit of the slope of the tangent line to the function at a given point. When the function is increasing at this point, the slope of the tangent line will be positive, and when the function is decreasing, the slope of the tangent line will be negative.
Similarly, if we take the second derivative of f(x), denoted as f”(x), we can determine whether the curvature of the function is concave upwards or downwards at x=2. This is because the sign of f”(x) will be positive if the curvature is concave upwards and negative if the curvature is concave downwards.
So, in summary, the given information tells us that the function f(x) is decreasing with a concave downwards curvature at x=2.
More Answers:
Unlocking The Local Maximum Point: Insights From Derivatives Of A Function At X=2Positive Slopes Of Derivatives: Explaining The Relationship Between Increasing Functions And Positive Slopes At A Point.
Increasing Functions: The Relationship Between F'(X) And F(X) At X=2