If f''(x)=-f(x) andg(x)=f'(x) andF(x)=fx22+gx22 andF(5)=5 then F(10)=?
5
10
0
15
Explanation for the correct option:
Find the value of F(10):
We have,
g(x)=f'(x)...(i)g'(x)=f''(x)=–f(x)...(ii)F(x)=fx22+gx22
Differentiate the above function with respect to x.
F'(x)=2fx2f'x212+2gx2g'x212[∵d(uv)dx=vu'+uv']=fx2f'x2+gx2g'x2=0
Since F'(x)=0 it means F(x)=constant
So F(5)=F(10)=5
Hence, the correct option is A.
Use the factor theorem to determine whether g(x) is a factor of f(x)
f(x)=22x2+5x+2;g(x)=x+2
Iff'(x)=g(x) and g'(x)=-f(x) for all x and f(5)=2=f'(5). Then f2(10)+g2(10) is