For every pair of continuous functions f, g : [0, 1] → ℝ such that f(x) : x [0, 1] = g(x) : x [0, 1] , the correct statement(s) is(are) :

JEE Advanced 2014 Mathematics question

For every pair of continuous functions f, g : [0, 1] → ℝ such that

\{f(x) : x \} = \{g(x) : x \},

the correct statement(s) is(are) :

(A) for some c ∈ [0, 1]Correct
(B) for some c ∈ [0, 1]
(C) for some c ∈ [0, 1]
(D) for some c ∈ [0, 1]Correct

Answer: (A),(D)

Solution

Let M be the common maximum, attained by f at and by g at . Then satisfies and ; by the intermediate value theorem h(c) = 0 for some c, i.e. f(c) = g(c).

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Question ID VK-026285