Let f(x) be a twice differentiable function such that f''(x) = -f(x) , f'(x) = g(x) and h(x) = f(x) ^2 + g(x) ^2 . If h(5) = 11 , then h(10) is equal to ...

Step-by-step solution

Given h(x) = (f(x))^2 + (g(x))^2 Differentiating with respect to x : h'(x) = 2f(x)f'(x) + 2g(x)g'(x) Given f'(x) = g(x) , differentiating gives f''(x) = g'(x) . Also given f''(x) = -f(x) , so g'(x) = -f(x) . Substituting these into h'(x) : h'(x) = 2f(x)g(x) + 2g(x)(-f(x)) = 0 Since h'(x) = 0 , h(x) is a constant function. Given h(5) = 11 , it follows that h(10) = 11 . Answer: 11