Mathsforum100 said:The second derivative at a maximum is either negative or zero. Can you explain how it can be zero? There can't be a 'plateau' at the maximum or it would not be a point. I cannot imagine graphically how the second derivative at a maximum can be zero. Before the maximum, the gradient is positive. After the maximum it is negative. So the gradient is decreasing.
The second derivative at a maximum is the rate of change of the slope of a function at its highest point. It is a mathematical tool used to determine whether a function has a maximum or minimum value at a specific point.
The second derivative is used to find a maximum by analyzing its sign at a specific point. If the second derivative is positive, the function has a minimum value at that point. If the second derivative is negative, the function has a maximum value at that point.
No, a function cannot have a maximum without a second derivative. The second derivative is essential in determining the concavity of a function, which is necessary for identifying maximum and minimum points.
The second derivative is the derivative of the first derivative. This means that it represents the rate of change of the slope of the function. A positive second derivative indicates that the slope is increasing, while a negative second derivative indicates that the slope is decreasing.
A local maximum is the highest point on a specific interval of a function, while a global maximum is the highest point of the entire function. A global maximum can also be a local maximum, but a local maximum does not necessarily have to be a global maximum.