f”(x) is positive
f'(x) is increasing and f(x) is concave up U
If the second derivative f”(x) of a function f(x) is positive, it means that the graph of the function is concave up, or in other words, it is a U-shape.
This can be understood by considering the definition of the second derivative. The second derivative tells us the rate at which the slope of the function is changing. If f”(x) is positive, this means that the slope of the function is increasing at a constant rate.
In terms of the graph, this means that the function is bending upwards, becoming steeper and steeper. Examples of this type of function include quadratic and cubic functions that open upwards.
One important consequence of the second derivative being positive is that the function has a minimum point at that point where the second derivative is positive. This means that the function starts decreasing after this point.
Therefore, f(x) has a local minimum at point x where f”(x) is positive. It should be noted that the function can continue to increase or it can start decreasing after reaching the minimum point.
More Answers:
The Relationship Between Second Derivative And Critical Points In Calculus.Concavity And The Second Derivative For Function F(X)
The Positive Second Derivative And Its Graphical Interpretation For U-Shaped Functions