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JEE Main2010MathematicsContinuity and DifferentiabilityActual

Let f: R R be a continuous function defined by f(x)= 1 e^x+2 e^ -x . Statement-1: f(c)= 1 3 , for some c R . Statement-2: 0 < f(x) 1 2 2 , for all x R

Options

  1. AStatement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1
  2. BStatement-1 is true, Statement-2 is false
  3. CStatement-1 is false, Statement-2 is true
  4. DStatement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

Correct answer

D. Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

Step-by-step solution

f(x)= 1 e^x+2 e^ -x = e^x e^ 2 x +2 f^ (x)= (e^ 2 x +2 ) e^x-2 e^ 2 x e^x (e^ 2 x+2 )^2 f^ (x)=0 e^ 2 x +2=2 e^ 2 x e ^ 2 x =2 e ^ x = 2 maximum f(x)= 2 4 = 1 2 2 0 < f(x) 1 2 2 x R Since 0 < 1 3 < 1 2 2 for some c Rf(c)= 1 3

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