If F (x)= ( f ( x 2 ) )^2+ ( g ( x 2 ) )^2 , where f ^ (x)=- f (x) and g(x)= f ^ (x) and given by F (5)=5 , then F (10) is equal to

Step-by-step solution

aligned & F (x)= ( f ( x 2 ) )^2+ ( g ( x 2 ) )^2 & ~F ^ (x)= 2 f ( x 2 ) f ^ ( x 2 ) 1 2 +2 ~g ( x 2 ) g ^ ( x 2 ) 1 2 &= f ( x 2 ) f ^ ( x 2 )+ g ( x 2 ) g ^ ( x 2 ) &= f ( x 2 ) g ( x 2 )+ g ( x 2 ) f ^ ( x 2 ) & [ g (x)= f ^ (x) g ^ (x)= f ^ (x) ] &= f ( x 2 ) g ( x 2 )+ g ( x 2 ) (- f ( x 2 ) ) &= 0 aligned F (x) is a constant for all x aligned & F(5)=5 & F(x)=5 for all x & F(10)=5 aligned