Chapter 4 | Applications of Derivatives
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a local extremum, but instead refer to that as an endpoint extremum. Given the graph of a function f , it is sometimes easy to see where a local maximum or local minimum occurs. However, it is not always easy to see, since the interesting features on the graph of a function may not be visible because they occur at a very small scale. Also, we may not have a graph of the function. In these cases, how can we use a formula for a function to determine where these extrema occur? To answer this question, let’s look at Figure 4.14 again. The local extrema occur at x =0, x =1, and x =2. Notice that at x =0 and x =1, the derivative f ′( x ) =0. At x =2, the derivative f ′( x ) does not exist, since the function f has a corner there. In fact, if f has a local extremum at a point x = c , the derivative f ′( c ) must satisfy one of the following conditions: either f ′( c ) =0 or f ′( c ) is undefined. Such a value c is known as a critical point and it is important in finding extreme values for functions. Definition Let c be an interior point in the domain of f . We say that c is a critical point of f if f ′( c ) =0 or f ′( c ) is undefined.
As mentioned earlier, if f has a local extremum at a point x = c , then c must be a critical point of f . This fact is known as Fermat’s theorem .
Theorem 4.2: Fermat’s Theorem If f has a local extremum at c and f is differentiable at c , then f ′( c ) =0.
Proof Suppose f has a local extremum at c and f is differentiable at c . We need to show that f ′( c ) =0. To do this, we will show that f ′( c ) ≥0 and f ′( c ) ≤0, and therefore f ′( c ) =0. Since f has a local extremum at c , f has a local maximum or local minimum at c . Suppose f has a local maximum at c . The case in which f has a local minimum at c can be handled similarly. There then exists an open interval I such that f ( c ) ≥ f ( x ) for all x ∈ I . Since f is differentiable at c , from the definition of the derivative, we know that f ′( c ) = lim x → c f ( x )− f ( c ) x − c . Since this limit exists, both one-sided limits also exist and equal f ′( c ). Therefore, (4.4) f ′( c ) = lim x → c + f ( x )− f ( c ) x − c , and (4.5) f ′( c ) = lim x → c − f ( x )− f ( c ) x − c . Since f ( c ) is a local maximum, we see that f ( x )− f ( c ) ≤0 for x near c . Therefore, for x near c , but x > c , we have f ( x )− f ( c ) x − c ≤0. From Equation 4.4 we conclude that f ′( c ) ≤0. Similarly, it can be shown that f ′( c ) ≥0. Therefore, f ′( c ) =0. □ From Fermat’s theorem, we conclude that if f has a local extremum at c , then either f ′( c ) =0 or f ′( c ) is undefined. In other words, local extrema can only occur at critical points.
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