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If f(x(1)-x(2)),f(x(1))f(x(2)) & f(x(1)+...

If `f(x_(1)-x_(2)),f(x_(1))f(x_(2))` & `f(x_(1)+x_(2))` are in A.P. for all `x_(1),x_(2)` and `f(0)!=0` then

A

`f^(')(5)=f(-5)`

B

`f(5)=f(-5)`

C

`f^(')(5)-f^(')(-5)=0`

D

`f^(')(5)+f^(')(-5)=0`

Text Solution

Verified by Experts

The correct Answer is:
B, D
`f(x_(1)+x_(2))+f(x_(1)-x_(2))=2f(x_(1)).f(x_(2))AA x_(1),x_(2)`
on putting `x_(1)=0, x_(2)=0` in above equation `f(0)+f(0)=2(f(0))^(2)`
`f(0)=1`
On putting `x_(1)=0,x_(2)=x_(1)` in equation i we get `f(x_(1))+f(-x_(1))=0impliesf^(')(5)+(f^(')(-5)=0`
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