If $\mathrm{f}\text{'}\left(\mathrm{x}\right) = \mathrm{g}\left(\mathrm{x}\right) \mathrm{and} {\mathrm{g}}^1\left(\mathrm{x}\right) = -\mathrm{f}\left(\mathrm{x}\right) \forall \mathrm{x} \mathrm{and} \mathrm{f}\left(2\right) = 4 = {\mathrm{f}}^1\left(2\right), \mathrm{then} {\mathrm{f}}^2\left(24\right) + {\mathrm{g}}^2\left(24\right) \mathrm{is}$

$$\begin{gathered} {\mathrm{f}}^1\left(\mathrm{x}\right)=\mathrm{g}\left(\mathrm{x}\right), {\mathrm{g}}^1\left(\mathrm{x}\right)=-\mathrm{f}\left(\mathrm{x}\right) \\ f\left(x\right)f\text{'}\left(x\right)+g\left(x\right)g\text{'}\left(x\right)=0 \\ Integrating on both sides \\ {f}^2\left(x\right)+g^2\left(x\right)=c \end{gathered}$$
f and g are constant function 
f(2) = g(2) = 4

${\mathrm{f}}^2\left(24\right)+{\mathrm{g}}^2\left(24\right)=4^2+4^2=32$