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Let f(x) and g(x) be continuously differentiable functions such that f'(x) = g(x) and f''(x) = -f(x). Define h(x) = [f(x)]² + [g(x)]². If h(0) = 5, what is the value of h(10)?
- 10
- 5
- 15
- None of these
Correct answer: 5
Solution
h' = 2 f f' + 2 g g' = 2 f g + 2 g f'' = 2 f g + 2 g(-f) = 0, so h is constant. Therefore h(10) = h(0) = 5.
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