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If the functions f(x) and g(x) are conti...

If the functions f(x) and g(x) are continuous on [a,b] and differentiable on (a,b) then in the interval (a,b) the equation
`|{:(f'(x),f(a)),(g'(x),g(a)):}|=(1)/(a-b)=|{:(f(a),f(b)),(g(a),g(b)):}|`

A

has at least one root

B

has exactly one root

C

has at most one root

D

no root

Text Solution

Verified by Experts

The correct Answer is:
A
Consider the function ` phi(x)` given by
`phi(x)|{:(f(a),f(x)),(g(a),g(x)):}|`
Since f(x), and g(x) are continuous on [a,b] and differentiable on (a,b) Therefore,`phi (x)` is continuous on [a,b] and differentiable (a,b) . Consequecly, by Largrange's mean value theorem therer exists at leat on `c in (a,b)` such that
`phi'(c)=(phi(b)-phi(a))/(b-a)`
`rArr |{:(f(a),f(c)),(g(a),g'(c)):}|=(1)/(b-a)=|{:(f(a),f(b)),(g(a),g(b)):}|`
`rArr|{:(f'(c),f(a)),(g'(c),g(a)):}|=(1)/(b-a)=|{:(f(a),f(b)),(g(a),g(b)):}|`
Hence, the equation
`|{:(f'(c),f(a)),(g'(x),g(a)):}|=(1)/(b-a)=|{:(f(a),f(b)),(g(a),g(b)):}`
has at least root in (a,b)
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