Question

If f"(x) = – f(x) and g(x) = f'(x) and F(x) = |[f (x/2)]|² + |[g(x/2)]|²  and F(5) = 5 then F(10) =________

(a) 5

(b) 10

(c) 0

(d) 15

Answer 1

The correct option (a) 5  

Explanation:

f"(x) = – f(x)

g(x) = f'(x)

F(x) = [f(x/2)]² + [g(x/2)]²

F(x) = [f(x/2)]² + [f'(x/2)]²

∴ F'(x) = [2 ∙ f(x/2) ∙ f'(x/2) + 2f'(x/2) ∙ f"(x/2)](1/2)

 = f'(x/2) [f(x/2) + f"(x/2)]

= f'(x/2) [f(x/2) – (x/2)]

= 0

∵ F' (x) = 0

⇒ F(x) has to be constant

∴ F(5) = F(10) = 5