Let Sfx f x f x0 Is this closed under addition and scalar mu

Let S={f(x)| f \'\'(x)+ f \'(x)=0}. Is this closed under addition and scalar multiplication?

*So you need to check that given f(x) and g(x) satisfying that equation, whether f(x)_+g(x) and C*f(x) (for all real number C) satisfy the equation also. Prove or find a counter example.*

Solution

Let f(x) and g(x) are two functions such that f\'\'(x)+f(x)=0 and g\'\'(x)+g(x)=0

and H(x) = f(x)+g(x)

H\'(x) = f\'(x)+g\'(x)

H\'\'(x) = f\'\'(x)+g\'\'(x)

H\'(x)+H\'\'(x) = (f\'\'(x)+f\'(x)) + (g\'\'(x)+g\'(x)) = 0+0 = 0.

So It is closed under addition.

f(x) be the function such that f\'\'(x)+f(x)=0.

Let g(x) = cf(x)

g\'(x) = cf\'(x)

g\'\'(x) = cf\'\'(x)

g\'(x)+g\'\'(x) = c(f\'(x)+f\'\'(x)) = 0.

So it is closed under scalar multiplication.

Example:

f(x) = e^(-x)

f\'(x) = -e^-(x)

f\'\'(x) = e^(-x)

f\'(x)+f\'\'(x) = -e^(-x)+e^(-x) = 0.

cf(x) = ce^(-x)

cf\'(x) = -ce^(-x)

cf\'\'(x) = ce^(-x)

c(f\'\'(x)+f\'(x)) = 0.

So it is closed under scalar multiplication.