Intermediate Value Theorem

Intermediate Value Theorem If f is continuous on the closed interval [a, b] and k is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = k First, read the “hypotheses”, the statements that first must be true in order to use a theorem. Look for the ‘if’ statements. 1) f is continuous on the closed interval [a, b] 2) k is any number between f(a) and f(b),

IVT Continued Now, look for the conclusion of the theorem, the statement after the word “then”. there is at least one number c in [a, b] such that f(c) = k To use a theorem, you must first satisfy each of the hypotheses, then you may use the conclusion

f(a) pick k = 2 f(b) [ a ] b c

[ ] The function is not continuous

Prob 84, page 78: Use the IVT to verify that in the interval [0, 3] the function has a value of 0. 1) The function is a polynomial so it is continuous for all x 2) f(0) = 8 and f(3) = -1 , so k = 0 is between f(a) and f(b) Therefore, conclude that a value x = c exists between [0, 3] such that f(c) = 0. Note: the theorem does give the value of c, it only says one exists.

Intermediate Value Theorem