Question

If $f\left(x\right),$ $\varphi \left(x\right),\phi \left(x\right)$ are continuous on [a, b] and differentiable on $(a,b)\exists cϵ(a,b)$, then

$\left|\begin{array}{ccc}f\left(a\right)& \varphi \left(a\right)& \phi \left(a\right)\\ f\left(b\right)& \varphi \left(b\right)& \phi \left(b\right)\\ f^{\prime }\left(c\right)& {\varphi }^{\prime }\left(c\right)& {\phi }^{\prime }\left(c\right)\end{array}\right|=$

• A. $f^{\prime }\left(c\right)$
• B. ${\varphi }^{\prime }\left(c\right)$
• C. ${\phi }^{\prime }\left(c\right)$
• D. $0$

Answer 1

The correct option is D $0$

Apply
Langranges mean value Theorem
$f^{\prime }\left(c\right)=\frac{f\left(a\right)-f\left(b\right)}{a-b}$ where $cϵ(a,b)$
${\varphi }^{\prime }\left(c\right)=\frac{\varphi \left(a\right)-\varphi \left(b\right)}{a-b}$
${\phi }^{\prime }\left(c\right)=\frac{\phi \left(a\right)-\phi \left(b\right)}{a-b}$
So $R_1\to R_1-R_2$

$\left|\begin{array}{ccc}f\left(a\right)-f\left(b\right)& \varphi \left(a\right)-\varphi \left(b\right)& \phi \left(a\right)-\phi \left(b\right)\\ f\left(b\right)& \varphi \left(b\right)& \phi \left(c\right)\\ f^{\prime }\left(c\right)& {\varphi }^{\prime }\left(c\right)& {\phi }^{\prime }\left(c\right)\end{array}\right|$
Dividing First Row Elements by $(a-b)$, we get,
$\left|\begin{array}{ccc}{f}^{\prime }\left(c\right)& {\varphi }^{\prime }\left(c\right)& {\phi }^{\prime }\left(c\right)\\ f\left(b\right)& \varphi \left(b\right)& \phi \left(b\right)\\ f^{\prime }\left(c\right)& {\varphi }^{\prime }\left(c\right)& {\phi }^{\prime }\left(c\right)\end{array}\right|=0$

Since, rows $1$ and $3$ are identical.