Project ForgeA function f(x) is continuous in the interval [0,2]. It is known that f(0) = f(2) = −1 and f(1) = 1. Which one of the following statements must be true?
- There exists a y in the interval (0,1) such that f(y) = f(y+1)
- BFor every y in the interval (0,1), f(y) = f(2−y)
- CThe maximum value of the function in the interval (0,2) is 1
- DThere exists a y in the interval (0,1) such that f(y) = −f(2−y)
Explanation
Define g(y) = f(y) − f(y+1) for y ∈ [0,1] (valid since y+1 ranges over [1,2], within f's domain). g(0) = f(0)−f(1) = −1−1 = −2, and g(1) = f(1)−f(2) = 1−(−1) = 2. Since g is continuous and changes sign across [0,1], the Intermediate Value Theorem guarantees some y in the OPEN interval (0,1) with g(y)=0, i.e. f(y)=f(y+1) — matching the file's own answer key. (The other options aren't guaranteed: B and C claim specific exact behavior the given data doesn't pin down.)
Source: GATE CS 2014 (Session 1) (2014)
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