5.2 Extreme Value Theorem, Global vs Local Extrema, and Critical Points

The Extreme Value Theorem states that if a function f is continuous on a closed interval [a, b], then f must have both a maximum and a minimum value on that interval. This means that for any function that is continuous on a closed interval, there will always be a highest and lowest point on that interval. One important concept to understand in relation to the Extreme Value Theorem is the difference between global extrema and local extrema. A global extremum refers to the highest or lowest point of a function on the entire interval, while a local extremum refers to the highest or lowest point of a function in a specific, smaller interval within the larger interval. For example, consider the function f(x) = x^2. The global extrema of this function would be the highest and lowest point on the entire interval, which in this case would be f(0) = 0 (the minimum) and f(1) = 1 (the maximum). However, if we consider the interval [-1,1], the local extrema would be f(-1) = 1 (the maximum) and f(1) = 1 (the minimum). Another important concept in relation to the Extreme Value Theorem is critical points. A critical point of a function is a point where the function is not differentiable, or where the derivative of the function is equal to 0. For example, in the function f(x) = x^2, the critical point is at x = 0, because the derivative of the function is equal to 0 at that point.

For the function f(x) = x^3, the global extrema are at x = -1 and x = 1, with f(-1) = -1 and f(1) = 1. The critical point of this function is at x = 0, where the derivative is equal to 0.

For the function f(x) = x^2 - 2x, the global extrema are at x = 0 and x = 2, with f(0) = 0 and f(2) = 2. The critical point of this function is at x = 1, where the derivative is equal to 0.

For the function f(x) = 1/x, the global extrema are at x = -1 and x = 1, with f(-1) = -1 and f(1) = 1. The critical point of this function is at x = 0, where the function is not differentiable.

For the function f(x) = sin(x), the global extrema are at x = pi/2 and x = 3pi/2, with f(pi/2) = 1 and f(3pi/2) = -1. The critical points of this function are at x = pi and x = 2pi, where the derivative is equal to 0.

For the function f(x) = e^x, the global extrema are at x = -infinity and x = infinity, with f(-infinity) = 0 and f(infinity) = infinity. The critical points of this function are at x = all real numbers, where the derivative is equal to e^x.

For the function f(x) = (x-2)^2, the global extrema are at x = 2 with f(2) = 0. The critical points of this function are at x = 2, where the derivative is equal to 0.

In addition to the concepts of global extrema and local extrema, and critical points, there are also the concepts of relative extrema and absolute extrema. A relative extremum is a point that is either a local maximum or a local minimum, while an absolute extremum is a point that is either a global maximum or a global minimum.

For example, consider the function f(x) = x^2. The global extrema of this function would be the highest and lowest point on the entire interval, which in this case would be f(0) = 0 (the minimum) and f(1) = 1 (the maximum). These are absolute extrema. However, if we consider the interval [-1,1], the local extrema would be f(-1) = 1 (the maximum) and f(1) = 1 (the minimum). These are relative extrema.

Another way to identify relative extrema is by looking at the first and second derivatives of a function. If the first derivative of a function is positive at a point and becomes negative as you move away from that point, then that point is a local maximum. If the first derivative of a function is negative at a point and becomes positive as you move away from that point, then that point is a local minimum. If the first derivative of a function is equal to 0 at a point and the second derivative is positive, then that point is a local minimum. If the first derivative of a function is equal to 0 at a point and the second derivative is negative, then that point is a local maximum.

Examples: For the function f(x) = x^4, the global extrema are at x = -1 and x = 1, with f(-1) = 1 and f(1) = 1. The critical points of this function are at x = -1 and x = 1, where the first derivative is equal to 0 and the second derivative is positive, indicating a local minimum.

For the function f(x) = -x^3, the global extrema are at x = -1 and x = 1, with f(-1) = -1 and f(1) = -1. The critical points of this function are at x = -1 and x = 1, where the first derivative is equal to 0 and the second derivative is negative, indicating a local maximum.

For the function f(x) = x^2 + x, the global extrema are at x = -1 and x = infinity, with f(-1) = -1 and f(infinity) = infinity. The critical points of this function are at x = -1/2, where the first derivative is equal to 0 and the second derivative is positive, indicating a local minimum.

It is important to note that for functions defined on a closed interval, the Extreme Value Theorem guarantees that there will be at least one global maximum and one global minimum. However, for functions defined on an open interval or on an unbounded domain, the Extreme Value Theorem does not necessarily apply and it may not be possible to find global extrema.

When comparing global and local extrema, it is important to keep in mind the context of the problem or the interval being considered. Global extrema represent the highest and lowest points of a function on the entire interval, while local extrema represent the highest and lowest points of a function in a specific, smaller interval within the larger interval.

For example, if a problem is asking to find the maximum or minimum value of a function over a specific interval, such as [0,2], then local extrema would be appropriate to consider. However, if a problem is asking to find the overall maximum or minimum value of a function over an entire domain, such as all real numbers, then global extrema would be the appropriate concept to consider.

It is also important to note that a global extremum is also a local extremum, but a local extremum is not necessarily a global extremum. A function may have multiple local extrema, but only one global extremum.

In addition, when solving optimization problems, we often look for global extrema, as they are the points where the function is at its highest or lowest over the entire domain. However, when solving certain types of differential equations, local extrema may be important in understanding the behavior of the function in specific regions.

In conclusion, the Extreme Value Theorem states that if a function is continuous on a closed interval, then it must have both a maximum and a minimum value on that interval. Understanding the concepts of global extrema and local extrema, and critical points is essential in identifying the extrema of a function. Additionally, by looking at the first and second derivatives of a function, we can also identify relative extrema.