If `f: RvecRa n dg: RvecR` are two functions such that `f(x)+f^(x)=-xg(x)f^(prime)(x)a n dg(x)>0AAx in Rdot` Then the function `f^2(x)+f('(x))^2` has a maxima at `x=0` a minima at `x=0` a point of inflexion at `x=0` none of these

Options

A. a maxima at x =0
B. a minima at x =0
C. a point of inflexion at x =0
D. a point of inflexion at x =0

Answer

1

Explanation

`f(x)+f''(x)=-g(x)f(x)`
Let `h(x) =f^(2)(x)+(f'(x))^(2)`
`therefore h(x)=2(x)f'(x)+2f(x)f''(x)`
`=2f'(x)[-x]g(x)f'(x)`
`=-2x(f'(x))^(2)g(X)`
Thus x =0 is a point of maxima for h(x)