Let f(x)= _ y y (x^ 1 / y -1 ) , and 2022 f ( 1 x )+P f(x)=f (x^2 ) , then P=

Step-by-step solution

Given, f(x)= _ y y (x^ 1 / y -1 ) and 2022 f ( 1 x )+p f(x)=f (x^2 ) Now, f(x)= _ y (x^ 1 / y -1 ) (1 / y) On putting y=1 / t array r As y , then t 0 f(x)= _ t 0 x^t-1 t array Use L'Hospital's rule, aligned & f(x)= _ t 0 x^t( x)-0 (1) = (x) & f ( 1 x )= ( 1 x )=- (x) aligned Now, 2022 f ( 1 x )+P f(x)=f (x^2 ) aligned & 2022[- (x)]+P[ x]= (x^2 ) & -2022+P=2 P=2024 aligned