question_answer

Let f be a continuous and differentiable function on R satisfying \[f(-x)=f(x)\] and \[f(2+x)=f(2-x)\forall x\in R\] and \[f'(1)=-5\]. Then the value of \[\sum\limits_{r=0}^{100}{{{(-1)}^{r}}f'(r)}\] is equal to

A) \[0\]         

B)        \[-100\]                 

C) \[100\]                   

D)        \[200\]

Correct Answer: A

Solution :

     [a]  \[f(-x)=f(x)\] \[\Rightarrow \,\,\,\,\,f'(-x)=f'(x)\] \[f'(1)=-5\] \[\Rightarrow \,\,\,\,\,\,f'(-1)=5\] Also, \[f(2+x)=f(2-x)\] \[\Rightarrow \,\,\,\,\,\,\,\,\,f'(2+x)=-f'(2-x)\] \[\Rightarrow \,\,\,\,\,\,\,\,\,f'(3)=-f'(1)=5\] \[\therefore \,\,\,\,\sum\limits_{r=0}^{100}{{{(-1)}^{r}}\,f'(r)=f'(0)-f'(1)+f'(2)-f'(3)+}\] \[....-f'(99)+f'(100)=0\]