A critical point in an interval problem is a point where the derivative of a function is equal to zero or undefined. It can also be the endpoints of the interval.
To find the critical points of an interval problem, you need to take the derivative of the function and set it equal to zero. Then, solve for the variable to find the x-values of the critical points. You also need to check the endpoints of the interval to see if they are critical points.
Critical points are important in interval problems because they can help determine the maximum and minimum values of a function within the given interval. They can also help identify any points of inflection or where the function changes from increasing to decreasing.
To determine if a critical point is a local maximum or minimum, you need to use the first or second derivative test. The first derivative test involves plugging in values to the first derivative on either side of the critical point. If the values are positive on one side and negative on the other, the critical point is a local maximum. If the values are negative on one side and positive on the other, the critical point is a local minimum. The second derivative test involves plugging in the critical point to the second derivative. If the value is positive, the critical point is a local minimum. If the value is negative, the critical point is a local maximum.
A critical point is a point where the derivative of a function is equal to zero or undefined. It can be a point of maximum, minimum, or neither. A point of inflection is a point where the concavity of a function changes. It is not necessarily a critical point, as the derivative can be defined at this point.