Local extrema are determined by finding critical points where the derivative of the function is zero or undefined, and then using the first or second derivative test to determine if the critical point is a local maximum or minimum. Global extrema are found by comparing the values of the function at the critical points, endpoints of the interval, and any other relevant points within the interval. The highest value among these points is the global maximum, and the lowest value is the global minimum. **

The local rate of change can be determined by calculating the derivative of a function at a specific point. This derivative represents the rate at which the function is changing at that particular point. By finding the slope of the tangent line to the curve at that point, we can determine the local rate of change. This can be done using calculus techniques such as differentiation. **

The local rate of change can be determined by calculating the derivative of a function at a specific point. This can be done using calculus techniques such as finding the slope of the tangent line to the curve at that point. Another method is to calculate the average rate of change over a small interval around the point and then take the limit as the interval approaches zero. This will give the instantaneous rate of change at that point. **

To determine the third degree function equation from a local maximum and a local minimum, you can start by setting up a general third degree function in the form of f(x) = ax^3 + bx^2 + cx + d. Then, use the information about the local maximum and minimum to set up a system of equations. The local maximum and minimum will give you two points on the function, which you can use to solve for the coefficients a, b, c, and d. Once you have the values of the coefficients, you can plug them back into the general third degree function to obtain the specific equation that represents the function with the given local maximum and minimum. **

To determine the local slope using a table and limit calculation, you can first create a table with values of x approaching the given point where you want to find the slope. Then, calculate the corresponding y-values using the function. Next, find the average rate of change between the points on either side of the given point by dividing the change in y by the change in x. Finally, take the limit of this average rate of change as the interval approaches zero to find the local slope at the given point. **

To determine the local maximum and local minimum of a function using the second derivative, we can analyze the sign of the second derivative at critical points. If the second derivative is positive at a critical point, the function has a local minimum at that point. If the second derivative is negative at a critical point, the function has a local maximum at that point. To find inflection points using the second derivative, we can analyze the sign changes of the second derivative. If the second derivative changes sign at a point, then that point is an inflection point of the function. If the second derivative is positive before the point and negative after the point, the function has a concave up to concave down transition and vice versa. **

One can determine whether an extremum problem is a local minimum or maximum by using the second derivative test. If the second derivative at the critical point is positive, then the function has a local minimum at that point. If the second derivative is negative, then the function has a local maximum at that point. If the second derivative is zero, then the test is inconclusive and other methods, such as the first derivative test or analyzing the behavior of the function around the critical point, may be used to determine the nature of the extremum. **

To calculate the derivative in mathematics, you use the concept of limits to find the rate of change of a function at a specific point. The derivative represents the slope of the tangent line to the function at that point, which gives the local rate of change. You can find the derivative using various methods such as the power rule, product rule, quotient rule, or chain rule, depending on the complexity of the function. Once you have the derivative, you can evaluate it at a specific point to determine the local rate of change of the function at that point. **