1. Tardigrade
  2. Question
  3. Mathematics
  4. 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)=

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Q. 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 $h (10)=$

Continuity and Differentiability

Solution:

Since $f’(x) = g(x), f' (x) = g' (x)$
Put $f' (x) = - f(x)$. Hence $g' (x) = -f(x)$
we have $h' (x) = 2f(x) f’(x) + 2g(x) g' (x)$
$= 2[f(x) g(x) + g(x) [-f(x)]] = 2 [f(x) g(x) - f(x) g(x)] = 0$
$\therefore h(x) = C$, a constant
$\therefore h(0)= C i.e. C = 5$
$h(x) = 5$ for all $x$. Hence $h (10) = 5.$