Question

If $$f''(x)=-f(x)$$ and $$g(x)=f'(x)$$ and $$\displaystyle{F}({x})=\left ({f}\left (\frac{{x}}{2}\right)\right)^{2}+\left ({g}\left (\frac{{x}}{2}\right)\right)^{2}$$ and given that $$F(5)=5$$, then $$F(10)$$ is equal to

A
5
B
10
C
0
D
15

Solution

The correct option is A $$5$$$$\mathrm{f}''(\mathrm{x})=-\mathrm{f}(\mathrm{x})$$ and $$\mathrm{f'}(\mathrm{x})=\mathrm{g}(\mathrm{x})$$ $$\Rightarrow \mathrm{f}''(\mathrm{x}).\ \mathrm{f'}(\mathrm{x})+\mathrm{f}(\mathrm{x}).\ \mathrm{f'}(\mathrm{x})=0$$$$Integrating,$$$$(\mathrm{f'}(\mathrm{x}))^{2}+(\mathrm{f}(\mathrm{x}))^{2}=\mathrm{c}\Rightarrow(\mathrm{f}(\mathrm{x})^{2}+(\mathrm{g}(\mathrm{x}))^{2}=\mathrm{c}$$$$\Rightarrow \mathrm{F}(\mathrm{x})=\mathrm{c}$$$$\Rightarrow \mathrm{F}(10)=5$$ Mathematics

Suggest Corrections

0

Similar questions
View More

People also searched for
View More