$ \begin{vmatrix} f(a) & g(a) & h(a) \\ f(b) & g(b) & h(b) \\ f'(c) & g'(c) & h'(c) \\ \end{vmatrix} =0 $

Question

If $f(x),g(x),h(x)$ have derivatives in $[a,b]$, show that there exists a value $c$ of $x$ in $(a,b)$ such that $$ \begin{vmatrix} f(a) & g(a) & h(a) \\ f(b) & g(b) & h(b) \\ f'(c) & g'(c) & h'(c) \\ \end{vmatrix} =0 $$

I am getting an idea of using generalized mean value theorem, but not able to proceed. Need help!

Answer 1

Let $$ F(x) = \det \begin{bmatrix} f(a) & g(a) & h(a) \\ f(b)& g(b) & h(b) \\ f(x) & g(x) & h(x) \end{bmatrix}, $$ then $F(a) = F(b )=0$. Also note that $$ F'(x) = \det \begin{bmatrix} f(a) & g(a) & h(a) \\ f(b)& g(b) & h(b) \\ f'(x) & g'(x) & h'(x) \end{bmatrix}, $$ which could be directly showed by definition of derivatives.

Answer 2

Hint: Consider the function

$$j(x)=\begin{vmatrix} f(a) & g(a) & h(a) \\ f(b) & g(b) & h(b) \\ f(x) & g(x) & h(x) \\ \end{vmatrix}.$$

Note that (assuming all functions are differentiable and thus continuous), $j$ is a continuous function with $j(a)=j(b)=0$...