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

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!

• Wikipedia – Robert Israel Oct 24 '18 at 15:11
• The functions are assumed to be at least $C^1$? – Chaos Oct 24 '18 at 15:15

## 2 Answers

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.

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 $$j(a)=j(b)=0$$...