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माना कि [0, 1] पर f, g संतत फलन इस तरह कि f(0) = f(1) = 0; g(0) = g(1) = 1 तथा f(1/2) > g(1/2)। निम्न कथनों में से कौन-सा सत्य है?

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  1. ऐसा कोई t ∈ [0, 1] नहीं है जिससे कि f(t) = g(t) हो
  2. वास्तव में एक t ∈ [0, 1] ऐसा है कि f(t) = g(t)
  3. कम से कम दो t ∈ [0, 1] ऐसे हैं कि f(t) = g(t)
  4. हमेशा अनंत रूप से कई t ∈ [0, 1] होते हैं जैसे कि f(t) = g(t)

Answer (Detailed Solution Below)

Option 3 : कम से कम दो t ∈ [0, 1] ऐसे हैं कि f(t) = g(t)
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व्याख्या:

मान लीजिये

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

समीकरण 1 और 2 से, एक t [0 , 1] है

समीकरण 2 और 3 से, एक t [ 0 ,1 ] है

इसलिए विकल्प (3) सही है

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