Applications of Derivatives. Extreme Values (Part II)

Applications of Derivatives. Extreme Values (Part II)

The critical points of a function represent the possible values of the function. A function has two sets of critical points: one for its x values and one for its y values. The x-critical points are defined as those points where the function is undefined. The y-critical points are defined as those points where the function is zero (constant). It is possible for a function to have some x-critical points and no y-critical points, or vice versa.

How can we know that a given function actually has absolute maximum or absolute minimum?

The first theorem we will look at is the Extreme Value Theorem. This theorem states that if f is a function that is continuous on a closed interval [a, b], then f always attains both an absolute maximum value and an absolute minimum value in the interval [a, b].

In the theorem here, the closed interval [a, b] is assumed. This is very important.

Get quality help now

Doctor Jennifer

Verified writer

Proficient in: Math

5 (893)

“ Thank you so much for accepting my assignment the night before it was due. I look forward to working with you moving forward ”

+84 relevant experts are online

Hire writer

If this interval is replaced by one of other types, then the theorem may not hold. That means that the function may not have an absolute maximum or an absolute minimum.

In the previous cases--with a, c, and d--absolute maximum or absolute minimum can be absent if the interval under consideration is not closed. Only when the interval is closed will both an absolute maximum and an absolute minimum exist.

Absolute maximum and absolute minimum are not guaranteed to exist for every interval of a function.

If we go back to the early examples we saw earlier, now in this case you see that the absolute maximum and the absolute minimum over the interval a, b may occur at an interior point or at an endpoint.

Both situations may occur. Further examples of this are illustrated by the three points illustrated. The absolute minimum of the function occurs at an interior point, as does its absolute maximum.

In the example shown here, the absolute minimum value is at an interior point while the absolute maximum values occur at the endpoints.

In this one, the absolute maximum occurs at the interior point and the absolute minimum occurs at the end point.

In both cases it can happen that the function has extreme values at interior points. How to locate extreme values? If the absolute maximum occurs at an interior point, then it is always situated at a peak. The tangent at such a point is horizontal, analogously for the absolute minimum.

Nevertheless, if the absolute maximum or minimum occurs at the end point, then the tangent need not be horizontal. The derivatives of functions can be used to locate the extreme points.

The First Derivative Theorem for Local Extreme Values states that, if f(x) attains a local maximum or minimum value at an interior point c, and the derivative of the function exists at c, then the derivative must be equal to zero.

f'(c)=0

Saying differently, if the function attains a maximum at an interior point and the tangent line exists, then the tangent has to be horizontal. So it is like here: This is the absolute maximum. You see the tangent is horizontal here. The proof goes as follows: Remember, the derivative at a point equals the limit when x tends to c of this ratio.

We consider the limit when x approaches c from both sides. The function f(x) attains a maximum at c, so f(x) - f(c) is always less than or equal to zero.

When x approaches c from the right-hand side, x minus c is positive. Thus, this ratio is negative. Therefore, the limit of the function f(x) as x approaches c from the right-hand side is less than or equal to zero. Similarly, since f(c) is a local maximum, f(x) minus f(c) is always less than or equal to zero.

As x approaches c from the left, x minus c becomes negative. Therefore, the whole ratio becomes greater than or equal to zero. To rephrase, when x tends to c minus , the limit is greater than or equal to zero. We assume f dash c exists; therefore these two limits must be equal to zero.

The first case tells us that the limit must be less than or equal to zero, and the second case tells us that it must be greater than or equal to zero. Therefore, the only possibility is that both limits are equal to zero. In other words, f dash c equals zero.

Hence, f'(c)=0

This is a proof that whenever you have an absolute maximum or a local maximum at the point c, then the tangent will be horizontal or is simply will not exist.

The consequence of Theorem 2 is as follows: where can f have extreme values? First, the extreme value may occur at the interior points where the derivative equals zero, or it may occur at an interior point such that f does not have a derivative there, or at the endpoints of the interval on which f is defined.

So, the following definition applies to interior points of an interval for a function f whose derivative is either zero or undefined. These points are called critical points.

Critical points are those points in the first two possibilities where either the derivative equals zero or where it does not exist.

The absolute extrema of a continuous function f, defined on a finite closed interval [a, b], can be found by first locating the critical points of f, which are those values where f'(x) = 0 or f'(x) is undefined. The derivative must also be evaluated at all critical points as well as the endpoints; sometimes the extreme values may occur at these points. If not the first two steps, we just have to compare all these values to find the largest and smallest of these values.