University:
Eastern Kentucky University
Course:
MAT 234 | Calculus I
Academic year:

2020
Views:

419

Pages:

1
Author:

Killian Mata

The Intermediate Value and the Extreme Value Theorems After discussing continuity at length, we will now learn two important consequences brought about by the continuity of a function over a closed interval. The first one is called the Intermediate Value Theorem or the IVT. The second one is called the Extreme Value Theorem or the EVT. The Intermediate Value Theorem The first theorem says that a function 𝑓(𝑥) which is found to be continuous over a closed interval [a, b] will take any value between 𝑓(𝑎) and 𝑓(𝑏). Theorem 4: Intermediate Value Theorem (IVT). If a function 𝑓(𝑥) is continuous over a closed interval [a, b], then for every value m between 𝑓(𝑎) and 𝑓(𝑏), there is a value 𝑐 ∈ [𝑎, 𝑏] such that 𝑓(𝑐) = 𝑚. Look at the graph as we consider values of m between 𝑓(𝑎) and 𝑓(𝑏). Imagine moving the dotted line for m up and down between the dotted lines for f(a) and f(b). Correspondingly, the dot P will move along the thickened curve between the two points, (a, f(a)) and (b, f(b)). We make the following observations: As the dark dot moves, so will the vertical dotted line over x = c move. In particular, the said line moves between the vertical dotted lines over x = a and x = b. More in particular, for any value that we assign m in between f(a) and f(b), the consequent position of the dark dot assigns a corresponding value of c between a and b. This illustrates what the IVT says. The Extreme Value Theorem The second theorem illustrates that a function 𝑓(𝑥) which is found to be continuous over a closed interval [a, b] is guaranteed to have extreme values in that interval. Theorem 5: Extreme Value Theorem (EVT). If a function f(x) is continuous over a closed interval [a, b], then 𝑓(𝑥) is guaranteed to reach a maximum and a minimum on [a, b]. An extreme value of 𝑓 , or extremum, is either a minimum or a maximum value of the function. A minimum value of 𝑓 occurs at some x = c if 𝑓(𝑐) ≤ 𝑓(𝑥) for all x ≠ c in the interval. A maximum value of 𝑓 occurs at some x = c if 𝑓(𝑐) ≥ 𝑓(𝑥) for all x ≠ c in the interval.

The Intermediate Value and the Extreme Value Theorems

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