Question

If the function f(x) and g(x) are continuous in [a, b] and differentiable in (a, b), then the equation $\left|\begin{array}{cc}f\left(a\right)& f\left(b\right)\\ g\left(a\right)& g\left(b\right)\end{array}\right|=(b-a)\left|\begin{array}{cc}f\left(a\right)& f^{\prime }\left(x\right)\\ g\left(a\right)& g^{\prime }\left(x\right)\end{array}\right|$ has, in the interval [a, b].

• A. Atleast one root
• B. Exactly one root
• C. Atmost one root
• D. No root

Answer 1

The correct option is A Atleast one root

Consider Lagrange's mean value theorem for $f\left(x\right)$ and $g\left(x\right)$ in $(b,a)$.

$\therefore f^{\prime }\left(x\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}\text{and}{g}^{\prime }\left(x\right)=\frac{g\left(b\right)-g\left(a\right)}{b-a}$ have atleast one real solution each.

Hence, a linear combination of these equations should also have atleast one real solution.

$\therefore f\left(a\right)g^{\prime }\left(x\right)-g\left(a\right)f^{\prime }\left(x\right)=f\left(a\right)\left(\frac{g\left(b\right)-g\left(a\right)}{b-a}\right)-g\left(a\right)\left(\frac{f\left(b\right)-f\left(a\right)}{b-a}\right)$

$\therefore (b-a)\left(f\right(a\left)g^{\prime }\right(x)-g(a\left)f^{\prime }\right(x\left)\right)=f\left(a\right)g\left(b\right)-f\left(a\right)g\left(a\right)-g\left(a\right)f\left(b\right)+g\left(a\right)f\left(a\right)$

$\therefore \left|\begin{array}{cc}f\left(a\right)& f\left(b\right)\\ g\left(a\right)& g\left(b\right)\end{array}\right|=(b-a)\left|\begin{array}{cc}f\left(a\right)& f^{\prime }\left(x\right)\\ g\left(a\right)& g^{\prime }\left(x\right)\end{array}\right|$

Hence, the above equation has atleast one root.