Question
If the functions $$f\left( x \right)$$ and $$g\left( x \right)$$ are continuous in $$\left[ {a,\,b} \right]$$ and differentiable in $$\left( {a,\,b} \right),$$ then equation \[\left| \begin{array}{l}
f\left( a \right)\,\,\,\,\,f\left( b \right)\\
g\left( a \right)\,\,\,\,\,g\left( b \right)
\end{array} \right| = \left( {b - a} \right)\left| \begin{array}{l}
f\left( a \right)\,\,\,\,\,f'\left( x \right)\\
g\left( a \right)\,\,\,\,\,g'\left( x \right)
\end{array} \right|\] has in the interval $$\left[ {a,\,b} \right]$$
A.
at least one root
B.
exactly one root
C.
at most one root
D.
no root
Answer :
at least one root
Solution :
Let \[h\left( x \right)\left| \begin{array}{l}
f\left( a \right)\,\,\,\,\,f\left( x \right)\\
g\left( a \right)\,\,\,\,\,g\left( x \right)
\end{array} \right| = f\left( a \right)g\left( x \right) - g\left( a \right)f\left( x \right)\]
Then, \[h'\left( x \right) = f\left( a \right)g'\left( x \right) - g\left( a \right)f'\left( x \right) = \left| \begin{array}{l}
f\left( a \right)\,\,\,\,\,f'\left( x \right)\\
g\left( a \right)\,\,\,\,\,g'\left( x \right)
\end{array} \right|\]
Since, $$f\left( x \right)$$ and $$g\left( x \right)$$ are continuous in $$\left[ {a,\,b} \right]$$ and differentiable in $$\left( {a,\,b} \right),$$ therefore $$h\left( x \right)$$ is also continuous $$\left[ {a,\,b} \right]$$ in and differentiable in $$\left( {a,\,b} \right).$$
So, by mean value theorem, there exists at least one real number $$c,\,a < c < b$$ for which $$h'\left( c \right) = \frac{{h\left( b \right) - h\left( a \right)}}{{b - a}},$$
$$\therefore \,h\left( b \right) - h\left( a \right) = \left( {b - a} \right)h'\left( c \right).....\left( {\text{i}} \right)$$
Here, \[h\left( a \right) = \left| \begin{array}{l}
f\left( a \right)\,\,\,\,\,f\left( a \right)\\
g\left( a \right)\,\,\,\,\,g\left( a \right)
\end{array} \right| = 0,\,\,h\left( b \right) = \left| \begin{array}{l}
f\left( a \right)\,\,\,\,\,f\left( b \right)\\
g\left( a \right)\,\,\,\,\,g\left( b \right)
\end{array} \right|\]
$$\therefore $$ From equation \[\left( {\rm{i}} \right),\,\left| \begin{array}{l}
f\left( a \right)\,\,\,\,\,f\left( b \right)\\
g\left( a \right)\,\,\,\,\,g\left( b \right)
\end{array} \right| = \left( {b - a} \right),\,\,h'\left( c \right) = \left( {b - a} \right)\left| \begin{array}{l}
f\left( a \right)\,\,\,\,\,f'\left( c \right)\\
g\left( a \right)\,\,\,\,\,g'\left( c \right)
\end{array} \right|\]