Question

Let f be a twice differentiable function such that f"x) = – f(x), f'(x) = g(x) and h(x) = [f(x)]² + [g(x)]², find h(10) if h(5) = 11

Answer 1

We have, h(x) = [f(x)]² + [g(x)]²

⇒ h'(x) = 2f(x)f'(x) + 2g(x)g'(x)

⇒ h'(x) = 2f(x)g(x) + 2g(x)g'(x)

⇒ h'(x) = – 2f"(x)g(x) + 2g(x)f''(x) = 0

Thus, h(x) is a constant function.

But h(5) = 11

⇒ h(x) = 11

⇒ h(10) = 11