Question

Let f : R → R be a continuous function defined
by f(x) = 1/e^x + 2e^-x
Statement - 1: f(c) = 1/3, for some c ∈ R.
Statement-2: 0 < f(x)≤ $fraction numerator 1 over denominator 2 square root of 2 end fraction$ , for all x ∈ R.

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

Statement-1 is true, Statement-2 is true; statement-2 is not a correct explanation for Statement-1.

Statement-1 is true, Statement-2 is false.

Statement-1 is false, Statement-2 is true.

Solution

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

because space straight f space left parenthesis straight x right parenthesis space equals space fraction numerator 1 over denominator straight e to the power of straight x space plus space begin display style 2 over straight e to the power of straight x end style end fraction Using comma space AM space greater or equal than space GM fraction numerator straight e to the power of straight x space plus space begin display style 2 over straight e to the power of straight x end style over denominator 2 end fraction space greater or equal than space open parentheses straight e to the power of straight x. space 2 over straight e to the power of straight x close parentheses to the power of 1 divided by 2 end exponent comma space as space straight e to the power of straight x greater than 0 rightwards double arrow space straight e to the power of straight x space plus space 2 over straight e to the power of straight x space greater or equal than space 2 square root of 2 rightwards double arrow space 0 less than space fraction numerator 1 over denominator straight e to the power of straight x space plus begin display style 2 over straight e to the power of straight x end style end fraction space less or equal than space fraction numerator 1 over denominator 2 square root of 2 end fraction therefore space 0 less than straight f left parenthesis straight x right parenthesis space less or equal than space fraction numerator 1 over denominator 2 square root of 2 end fraction comma space for space all space straight x element of straight R

Statement I us true and statement I as for some 'c'

f(c) = 1/3

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