There must be a typographical error in the question: the first line ##x < 2 \hspace{2em} y'>0, y''<0## should read as ##x < - 2 \hspace{2em} y'>0, y''<0##.Fatima Hasan said:
A twice-differentiable function is a function that can be differentiated twice. This means that its graph will have a well-defined curvature, which can provide information about its behavior and rate of change.
A function is twice-differentiable if it has continuous derivatives up to the second order. This means that the first and second derivatives must exist and be continuous at every point in the function's domain.
The graph of a twice-differentiable function can have various shapes and forms, depending on the specific function. However, it will always have a smooth and continuous curve with a well-defined curvature.
Yes, it is possible for a function to be twice-differentiable at some points and not at others. This can occur if the function has a sharp corner or a discontinuity at a certain point, where its derivatives do not exist.
The second derivative of a function can provide information about the concavity and inflection points of its graph. A positive second derivative indicates a concave-upward curve, while a negative second derivative indicates a concave-downward curve. Inflection points are points where the curvature of the graph changes from concave-upward to concave-downward or vice versa.
