If $f''\left( x \right) = - f\left( x \right)$, where $f\left( x \right)$ is a continuous double differentiable function and ${\rm{g}}\left( x \right) = f'\left( x \right).\,$ If $F\left( x \right) = {\left( {f\left( {\frac{x}{2}} \right)} \right)^2} + {\left( {{\rm{g}}\left( {\frac{x}{2}} \right)} \right)^2}$ and $F\left( 5 \right) = 5$ then $F\left( {10} \right)$

JEE MATHEMATICS MCQ: DIFFERENTIATION

1 Medium

1 of 1

Analyzing +4 | -1

f''\left( x \right) = - f\left( x \right)

, where

f\left( x \right)

is a continuous double differentiable function and

\mathrm{g}\left( x \right) = f'\left( x \right).\,

F\left( x \right) = {\left( {f\left( {\frac{x}{2}} \right)} \right)^2} + {\left( {\mathrm{g}\left( {\frac{x}{2}} \right)} \right)^2}

and

F\left( 5 \right) = 5

then

F\left( {10} \right)

View answer and explanation

Correct option: B

Option B (5) is correct because

F'(x) = 0

, indicating that

F(x)

is a constant function. Therefore,

F(10) = F(5) = 5