goldfish9776 said:
A critical point is a point on a graph or function where the slope (or derivative) is equal to zero. This means that the rate of change is neither increasing nor decreasing, and the function may have a local maximum, minimum, or saddle point at this point.
A stationary point is a point on a graph or function where the slope (or derivative) is equal to zero. This is the same definition as a critical point. However, the term "stationary point" is often used in optimization problems, while "critical point" is used more generally in calculus.
There is no difference between a critical point and a stationary point. Both terms refer to a point on a graph or function where the slope (or derivative) is equal to zero. The difference lies in their usage, with "critical point" being a more general term and "stationary point" being used in optimization problems.
To determine the type of critical/stationary point, you can use the second derivative test or the first derivative test. The second derivative test involves evaluating the second derivative at the critical/stationary point. If the second derivative is positive, the point is a local minimum. If the second derivative is negative, the point is a local maximum. If the second derivative is zero, the test is inconclusive and you may need to use the first derivative test. The first derivative test involves evaluating the first derivative at points close to the critical/stationary point. If the first derivative changes from negative to positive, the point is a local minimum. If the first derivative changes from positive to negative, the point is a local maximum. If the first derivative does not change sign, the point is a saddle point.
Critical/stationary points are used in optimization problems to find the maximum or minimum value of a function. By finding the critical/stationary points and determining their type (using the second derivative test or first derivative test), we can determine the maximum or minimum value of the function. This is useful in various fields such as economics, engineering, and physics.
