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A function f(x) satisfies f''(x) = -f(x) and f'(x) = g(x). Define h(x) = [f(x)]² + [g(x)]². Given h(5) = 5, find h(10).

1. 0
2. 5
3. 10
4. 15

Correct answer: 5

Solution

h(x) = [f(x)]² + [g(x)]². h'(x) = 2*f(x)*f'(x) + 2*g(x)*g'(x). Now f'(x) = g(x) and g'(x) = f''(x) = -f(x). So h'(x) = 2*f(x)*g(x) + 2*g(x)*(-f(x)) = 2*f*g - 2*f*g = 0. Therefore h(x) is constant for all x. Since h(5) = 5, h(10) = 5.

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