# Let \(\phi(x)=\left\|\begin{array}{rcl}f(x)&g(x)&h(x)\\-f(a)&g(a)&h(a)\\f(b)&g(b)&h(b)\end{array}\right|\), \(g(x)=x\) and \(h(x)\) for all \(x\ina,b\), to deduce the Lagrange’s mean value theorem from the Generalized mean value theorem we must obtain the determinant

Let \(\phi(x)=\left\|\begin{array}{rcl}f(x)&g(x)&h(x)\\-f(a)&g(a)&h(a)\\f(b)&g(b)&h(b)\end{array}\right|\), \(g(x)=x\) and \(h(x)\) for all \(x\in[a,b]\), to deduce the Lagrange’s mean value theorem from the Generalized mean value theorem we must obtain the determinant

\(\left\|\begin{array}{rcl}1&g'(x)&0\\f(a)&a&1\\f(b)&a&1\end{array}\right|=0\)

\(\left\|\begin{array}{rcl}f'(x)&g'(x)&0\\f(a)&a&1\\f(b)&b&1\end{array}\right|=0\)

\(\left\|\begin{array}{rcl}f'(x)&g'(x)&0\\f(a)&g(a)&1\\f(b)&g(b)&1\end{array}\right|=0\)

—>> \(\left\|\begin{array}{rcl}f'(x)&1&0\\f(a)&a&1\\f(b)&b&1\end{array}\right|=0\)