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Given f, g are continuous functions on [0, 1] such that f(0) = f(1) = 0; g(0) = g(1) = 1 and f(1/2) > g(1/2). Which of the following statements is true?

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CSIR-UGC (NET) Mathematical Science: Held on (26 Nov 2020)
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  1. There is no t ∈ [0, 1] such that f(t) = g(t)
  2. There is exactly one t ∈ [0, 1] such that f(t) = g(t)
  3. There are at least two t ∈ [0, 1] such that  f(t) = g(t)
  4. There are always infinitely many t ∈ [0, 1] such that  f(t) = g(t)

Answer (Detailed Solution Below)

Option 3 : There are at least two t ∈ [0, 1] such that  f(t) = g(t)
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Detailed Solution

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Explanation:

Let 

h(x) = f(x) - g(x) 

h(0) = f(0) - g(0) = -1 < 0 . . . . . . . . . 1

h(1/2) = f(1/2) - g(1/2) > 0   . . . . . . . . 2

h(1) = f(1) - g(1) = -1 < 0   . . . . . . .  3

by equation 1 and 2 , there is one t \(\in\) [0 , 1]

by equation 2 and 3 , there is one t \(\in\) [ 0 ,1 ]

Hence option (3) is correct 

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