Let f be a differentiable function such that f (1)=2 and f ^ (x)= f (x) , for all x R . If h (x)= f ( f (x)) , then h ^ (1) is equal to

Step-by-step solution

Given: f ^ (x)= f (x) for all x R ^ f ^ (x) f (x) =1 Integrating on both sides, wwe get | f (x)|=x+ c f (x)= e ^ x+c f (x)= e ^x e ^ c f (x)= e ^x c ₁ ... (i) [ e ^ c = c ₁ ] As f(1)=2 aligned & c ₁ e =2 & c ₁= 2 e aligned Equation (i) becomes f (x)= e ^x 2 e Now, h (x)= f ( f (x)) array ll h ^ (x)= f ^ ( f (x)) f ^ (x) h ^ (1)= f ^ ( f (1)) f ^ (1) h ^ (1)= f ^ (2) f ^ (1) h ^ (1)= e ^2 2 e 2 h ^ (1)=4 e array