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Maths-

General

Easy

Question

If $f open parentheses x close parentheses equals fraction numerator 1 over denominator x plus 1 end fraction minus log invisible function application left parenthesis 1 plus x right parenthesis comma blank x greater than 0$ , then $f$ is

an increasing function
a decreasing function
both increasing and decreasing function
None of the above

The correct answer is: a decreasing function

Given curve is $f open parentheses x close parentheses equals fraction numerator 1 over denominator x plus 1 end fraction minus log invisible function application open parentheses 1 plus x close parentheses$
On differentiating w.r.t. $x$ , we get

$f to the power of ´ end exponent open parentheses x close parentheses equals negative fraction numerator 1 over denominator open parentheses x plus 1 close parentheses to the power of 2 end exponent end fraction minus fraction numerator 1 over denominator 1 plus x end fraction$

$rightwards double arrow blank f to the power of ´ end exponent open parentheses x close parentheses equals negative open square brackets fraction numerator 1 over denominator x plus 1 end fraction plus fraction numerator 1 over denominator open parentheses x plus 1 close parentheses to the power of 2 end exponent end fraction close square brackets$

$rightwards double arrow blank f to the power of ´ end exponent open parentheses x close parentheses equals negative$ ve, when $x greater than 0$

$therefore blank f left parenthesis x right parenthesis$ is a decreasing function

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