If functions f(x) and g(x) are continuous in [a, b] and differentiable in (a, b) n which of following is correct if re exists at least one point c, a < c < b, such that \(\begin{vmatrix}f( a ) & f( b ) \\g( a ) & g( b ) \end {vmatrix}\)?

Let, F(x) = \(\begin{vmatrix}f( a ) & f( b ) \\g( a ) & g( b ) \end {vmatrix}\) = f( a )g(x) ? f(x)g( a ) ?..(1)
=> F?(x) = f?( a )g?(x) ? f?(x)g( a )
Since, f(x) and g(x) are continuous in [a, b] and differentiable in (a, b),
So, F(x) is continuous in [a, b] and differentiable in (a, b)
Also from (1), F( a ) = f( a )g( a ) ? f( a )g( a ) = 0
And F( b ) = f( a )g( b ) ? f( b )g( a )
Now, by the mean value theorem, there exists at least one point c, a < c < b, such that,
F?( c ) = (F?( b ) ? F?( a )) / (b ? a)
=> f( a ) g?( c ) ? g( a ) f?( c ) = (f( a )g( b ) ? f( b )g( a ) ? 0)/b ? a
Or, f( a )g( b ) ? f( b )g( a ) = (b ? a)( f( a ) g?( c ) ? g( a ) f?( c ))
=>\(\begin{vmatrix}f( a ) & f( b ) \\g( a ) & g( b ) \end {vmatrix}\) = (b ? a)\(\begin{vmatrix}f( a ) & f'( c ) \\g( a ) & g'( c ) \end {vmatrix}\)