GATE CS 2014 Set 1 — Question 57
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Engineering Mathematics → Calculus → Limits & Continuity
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Question
A function is continuous in the interval . It is known that and . Which one of the following statements must be true?
Correct answer
(A) There exists a y in the interval (0,1) such that f(y) = f(y+1)
Solution
Let . We want to find if for some .
Evaluate at the endpoints of the interval :Since is continuous, is also continuous. Because and , by the Intermediate Value Theorem (IVT), there must exist a value such that .Thus, statement (A) must be true.
Evaluate at the endpoints of the interval :Since is continuous, is also continuous. Because and , by the Intermediate Value Theorem (IVT), there must exist a value such that .Thus, statement (A) must be true.
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