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Maths-

General

Easy

Question

Let f x( ) and g x( ) are defined and differentiable for $x greater or equal than x subscript 0 end subscript$ and $f open parentheses x subscript 0 close parentheses equals g open parentheses x subscript 0 close parentheses comma f to the power of ´ left parenthesis x right parenthesis greater than straight g to the power of straight prime left parenthesis x right parenthesis text for end text x greater than x subscript 0$ then

$f (x) < g (x) for some x > x_{0}$
$f (x) = g (x) for some x > x_{0}$
$f (x) > g (x) for some x > x_{0}$
$f (x) > g (x) for some x < x_{0}$

The correct answer is: $f left parenthesis x right parenthesis greater than g left parenthesis x right parenthesis text for some end text x greater than x subscript 0 end subscript$

$ϕ left parenthesis x right parenthesis equals f left parenthesis x right parenthesis minus g left parenthesis x right parenthesis text where end text x element of open square brackets x subscript 0 end subscript comma b close square brackets$ by $c element of open parentheses x subscript 0 end subscript comma b close parentheses contains as member$
$ϕ to the power of ´ end exponent left parenthesis c right parenthesis equals fraction numerator ϕ left parenthesis b right parenthesis minus ϕ open parentheses x subscript 0 end subscript close parentheses over denominator b minus x subscript 0 end subscript end fraction greater than 0$
$therefore f left parenthesis x right parenthesis greater than g left parenthesis x right parenthesis text for end text x equals b$

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