Question

Let f(x), g(x) be two continuously differentiable functions

satisfying the relationships f¢(x) = g(x) and f¢¢(x) = – f(x).

Let h(x) = [f(x)]² + [g(x)]². If h(0) = 5, then h(10) =

• A. 10
• B. 5
• C. 15
• D. None of these

Answer 1

Answer: (B)

5

Answer 2

Let f(x), g(x) be two continuous differentiable functions

Since f¢(x) = g(x), f¢¢(x) = g¢(x)

Put f¢¢(x) = –f(x)

Hence g¢(x) = –f(x)

We have h¢(x) = 2f(x) f¢(x) + 2g(x) g¢(x)

= 2[f(x) g(x) + g(x) [–f(x)]]

= 2[f(x) g(x) – f(x) g(x)] = 0

\ h(x) = C, a constant

\ h(0) = C i.e. C = 5

h(x) = 5 for all x.

Hence h(10) = 5.