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Most Important Selected Qs for JEE AdvancedMathematicsApplication of Derivatives

Let f:(0, ) R be a differentiable function satisfying f(x)+e^ f(x) = 2 x - x-1 . Find the number of integers in the range of x satisfying the inequality f (2 x^2+1 )-f (x^2+5 ) f(1), x>0 .

Correct answer

2

Step-by-step solution

f(x)+e^ f(x) = 2 x - x-1 ...(1) Differentiate both sides f^ (x)+e^ f(x) f^ (x)= -2 x^2 - 1 x 0 Hence, f^ (x) 0 f is decreasing Put x=1 in equation (1) f(1)+e^ f(1) =1 aligned & f(1)=0 & f (2 x^2+1 )-f (x^2+5 ) f(1), f(1)=0 aligned f (2 x^2+1 ) f (x^2+5 ), f is decreasing aligned 2 x^2+1 & x^2+5 x^2 & 4 -2 & x 2, but f:(0, ) R aligned so the x that satisfies the inequality belongs to 0 < x 2

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