Whether the statement, “If f is continuous on and and , then there exists a number r such that and ” is true or false.
The statement is false.
Theorem used: The Intermediate value Theorem
Suppose that if f is continuous on the closed interval [a, b] and let N be any number between and , where . Then there exists a number c in (a, b) such that .
Suppose f is continuous on and and .
Let lies between 3 and 4. That is, .
Then by Intermediate value theorem states that, there is a number r in the interval (−1, 1) such that .
Thus, there is a number r such that (or ) and .
Therefore, the given statement is true.
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