Question

if f(x+3) =f(x) +f(5)then prove that:f(2)=0 and f(5)=-f(-1)

Answer 1

Answer:

Putting x=0,y=5 in the given equation, we get

f(0+5)=f(0)f(5)

⟹f(5)[f(0)−1]=0

⟹f(0)=1

Consider, f

′

(5)=

h→0

lim

h

f(5+h)−f(5)

=

h→0

lim

h

f(5)f(h)−f(5)

=

h→0

lim

h

f(5)[f(h)−1]

=f(5)

h→0

lim

h

f(h)−f(0)

=f(5)f

′

(0)=2×3=6

∴f

′

(5)=6