The Relationship Between Critical Points and Turning Points in Mathematics | Explained

All turning points are _________ points but not all stationary points are ________

All turning points are critical points but not all stationary points are critical points

All turning points are critical points but not all stationary points are critical points.

A critical point in mathematics refers to a point where the derivative of a function either equals zero or is undefined. In other words, it’s a point where the function’s rate of change is changing, or where the slope or gradient is zero.

A turning point, also known as an extremum, refers to a point where a function changes from increasing to decreasing or from decreasing to increasing. It is the point where the function reaches its maximum or minimum value. Turning points can be classified as either a maximum point (where the function reaches its highest value) or a minimum point (where the function reaches its lowest value).

Now, it is important to note that all turning points are critical points because these points mark a change in the function’s behavior. At a turning point, the derivative of the function is zero (horizontal tangent) or undefined (vertical tangent). However, not all critical points are turning points.

A stationary point is a point on a curve where the derivative is zero. These points can be either turning points or points of inflection (where the function is changing concavity but not reaching a maximum or minimum value). In other words, a stationary point can be a turning point, but it can also be a point of inflection.

In summary, all turning points are critical points because they mark a change in the behavior of the function, and the derivative is either zero or undefined at these points. On the other hand, not all stationary points are turning points because they can also be points of inflection.

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