If x=a is a root of multiplicity two of a polynomial equation f(x)=0 , then

Step-by-step solution

Given that, x=a is a root of multiplicity two of a polynomial equation f(x)=0 . Let f(x)=(x-a) g(x) On differentiating w.r.t. x , we get f^ (x)=2(x-a) g(x)+(x-a)^2 g^ (x) Again, differentiating w.r.t x , we get aligned & Now, aligned & f^ (x)=2 g(x)+2(x-a) g^ (x)+2(x-a) g^ (x) &+(x-a)^2 g^ (x) aligned & =2 g(x)+4(x-a) g^ (x)+(x-a)^2 g^ (x) aligned At x=a , aligned & f^ (a)=2(a-a) g(a)+(a-a)^2 g^ (a)=0 & f^ (a)=2 g(a)+4(a-a) g^ (a)+(a-a)^2 g^ (a) & =2 g(a) aligned Hence, f(a)=f^ (a)=0, f^ (a) 0