Unizor - Derivatives - Minimum or Maximum, or Inflection

Notes to a video lecture on http://www.unizor.com

Derivatives - MIN or MAX

We know that if a smooth function f(x), defined on interval (a,b) (finite or infinite) has a local maximum or minimum at some point x0, its derivative at this point equals to zero.

Consider the converse statement: if a derivative of a smooth function f(x), defined on interval (a,b), equals to zero at some point x0, it attains its local minimum or maximum at this point. Is it a correct statement?
The answer is NO.
Here is a simple example. Function f(x)=x³ is defined for all real arguments and is monotonically increasing, so it has no minimum and no maximum. Its derivative equals to f I(x) = 3x². It is non-negative, as the derivative of a monotonically increasing function should be, but, in particular, it is equal to zero at x=0.
So, a point where derivative equals to zero is not necessarily a point of minimum or maximum.

All points where a derivative equals to zero are called stationary points of a function. Some of them are points of local minimum, some - local maximum and others (not local minimums, nor maximums) are called inflection points.

Example of a local minimum is a function f(x) = x² at point x=0 with derivative f I(x) = 2x, which equals to zero at x=0.


Example of a local maximum is a function f(x) = −x² at point x=0 with derivative f I(x) = −2x, which equals to zero at x=0.


Example of a point of inflection is a function f(x) = x³ at point x=0 with derivative f I(x) = 3x², which equals to zero at x=0.


Our task now is to distinguish different kinds of stationary points of a smooth function, which ones are local minimums, which are maximums and which are inflection points.
To accomplish this, we need to analyze the behavior of both the first and the second derivatives of a function.

Examine the point of local minimum of a smooth function first. The fact that point x0 is a local minimum means that in the immediate neighborhood of this point the behavior of a function, as we increase the argument from some value on the left of x0 to some value on the right of x0, is to monotonically decrease its value, reaching minimum value at this point x0 and then to monotonically increase afterwards.
As we know, monotonically decreasing smooth functions have a non-positive derivative, while monotonically increasing - non-negative. Since in the immediate neighborhood of point x0 the only point where a derivative is equal to zero is only our point of local minimum x0, we conclude that a derivative to the left of a point of local minimum x0 is strictly negative and to the right of it - strictly positive. So, a derivative changes the sign from minus to plus going through a point of local minimum.
Therefore, our first tool to distinguish among different types of stationary points (those where a derivative equals to zero) is analyze the sign of a derivative to the immediate left and to the immediate right of a stationary point. If it changes from minus to plus, it's a point of local minimum.

In our first example above function f(x) = x² at stationary point x=0 has derivative f I(x) = 2x that changes the sign from minus to plus as we move from negative argument to positive over point x=0. That identifies stationary point x=0 as local minimum.

Analogously, it a derivative changes the sign from plus to minus in the immediate neighborhood of a stationary point, its a point of local maximum.

In our second example above function f(x) = −x² at stationary point x=0 has derivative f I(x) = −2x that changes the sign from plus to minus as we move from negative argument to positive over point x=0. That identifies stationary point x=0 as local maximum.

Finally, if a derivative does not change the sign, but, being negative on the left of stationary point, is increasing to zero at a stationary point and then goes negative again, it's an inflection point. Similarly, if a derivative does not change the sign, but, being positive on the left of stationary point, is decreasing to zero at a stationary point and then goes positive again, it's an inflection point as well.

In our third example above function f(x) = x³ at stationary point x=0 has derivative f I(x) = 3x² that does not change the sign as we move from negative argument to positive over point x=0. It's positive on both sides of a stationary point. That identifies stationary point x=0 as an inflection point.

Another approach to identify stationary points of a smooth function as local minimum, maximum or inflection points is to analyze the second derivative.

Assuming x0 is a stationary point of function f(x), that is f I(x0) = 0, let's check the function's second derivative at this point f II(x0). It can be positive, negative or zero.

If it's positive, it means that the first derivative (relative to which the second derivative of the original function is the first derivative) is monotonically increasing. Since the first derivative equals to zero at point x0, it must be negative to the left of this stationary point and positive to the right, that is it changes the sign from minus to plus and the stationary point is a local minimum.
In our first example of function f(x)=x² the second derivative is f II(x0) = 2 (constant), which at the stationary point x=0 equals to 2. It is positive, therefore we deal with local minimum.

If the second derivative at the stationary point is negative, it means that the first derivative (relative to which the second derivative of the original function is the first derivative) is monotonically decreasing. Since the first derivative equals to zero at point x0, it must be positive to the left of this stationary point and negative to the right, that is it changes the sign from plus to minus and the stationary point is a local maximum.
In our second example of function f(x)=−x² the second derivative is f II(x0) = −2 (constant), which at the stationary point x=0 equals to −2. It is negative, therefore we deal with local maximum.

Finally, if the second derivative is equal to zero at the stationary point, we cannot make any judgment looking on the value of the second derivative at the stationary point.
Consider function f(x)=x4. Its first derivative is f I(x0) = 4x³, it's equal to zero at x=0, so this is a stationary point. The second derivative is f II(x0) = 12x², which at the stationary point x=0 equals to zero. Since it's zero, we cannot identify this stationary point as local minimum, maximum or inflection, though visually it's a clear minimum. This illustrates that the method based on the value of a second derivative at the stationary point is not always working.