A turning point is a point at which the derivative changes sign. A turning point may be either a relative maximum or a relative minimum. If the function is differentiable, then a turning point is a stationary point; however not all stationary points are turning points. If the function is twice differentiable, the stationary points that are not turning points are horizontal inflection points. For example, the function has a stationary point at x=0, which is also an inflection point, but is not a turning point.
a rising point of inflection is one where the derivative of the function is positive on both sides of the stationary point; such a point marks a change in concavity;
a falling point of inflection is one where the derivative of the function is negative on both sides of the stationary point; such a point marks a change in concavity.
The first two options are collectively known as "local extrema". Similarly a point that is either a global maximum or a global minimum is called a global extremum. The last two options—stationary points that are not local extremum—are known as saddle points. By Fermat's theorem, global extrema must occur on the boundary or at stationary points.
Curve sketching
Determining the position and nature of stationary points aids in curve sketching of differentiable functions. Solving the equation f' = 0 returns the x-coordinates of all stationary points; the y-coordinates are trivially the function values at those x-coordinates. The specific nature of a stationary point at x can in some cases be determined by examining the second derivativef'':
If f'' < 0, the stationary point at x is concave down; a maximal extremum.
If f'' > 0, the stationary point at x is concave up; a minimal extremum.
If f'' = 0, the nature of the stationary point must be determined by way of other means, often by noting a sign change around that point.
A more straightforward way of determining the nature of a stationary point is by examining the function values between the stationary points. A simple example of a point of inflection is the function f = x3. There is a clear change of concavity about the point x = 0, and we can prove this by means of calculus. The second derivative of f is the everywhere-continuous 6x, and at x = 0, f′′ = 0, and the sign changes about this point. So x = 0 is a point of inflection. More generally, the stationary points of a real valued function are those points x0 where the derivative in every direction equals zero, or equivalently, the gradient is zero.
Example
For the function f = x4 we have f' = 0 and f = 0. Even though f = 0, this point is not a point of inflection. The reason is that the sign of f changes from negative to positive. For the function f = sin we have f' ≠ 0 and f = 0. But this is not a stationary point, rather it is a point of inflection. This is because the concavity changes from concave downwards to concave upwards and the sign of f does not change; it stays positive. For the function f = x3 we have f' = 0 and f = 0. This is both a stationary point and a point of inflection. This is because the concavity changes from concave downwards to concave upwards and the sign of f' does not change; it stays positive.
