For the function f(x)= x¹⁰⁰ 100 + x⁹⁹ 99 + x² 2 +x+1 f ^ (1)= mf ^ (0), where m is equal to

Step-by-step solution

Given, f(x)= x¹⁰⁰ 100 + x⁹⁹ 99 + + x² 2 +x+1 f^ (x)= 100 x⁹⁹ 100 + 99 x⁹⁸ 99 + + 2 x 2 +1+0 [ f ( x )= x ^ n f ^ ( x )= nx ^ n -1 ] f ^ ( x )= x ⁹⁹+ x ⁹⁸+ + x +1 ...(i) Putting x =1 , we get f^ (1)= (1)⁹⁹+1⁹⁸+ +1+1 _ 100 times = 1+1+1 +1+1 _ 100 times f ^ (1)=100 ...(ii) Again, putting x =0 , we get f ^ (0)=0+0+ +0+1 f ^ (0)=1 ...(iii) From eqs. (ii) and(iii), we get; f ^ (1)=100 f ^ (0) Hence, m=100