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Maths-

General

Easy

Question

The value of $x$ for which $1 plus x log subscript e end subscript invisible function application open parentheses x plus square root of x to the power of 2 end exponent plus 1 end root close parentheses greater or equal than square root of x to the power of 2 end exponent plus 1 end root a r e$

$x \leq 0$
$0 \leq x \leq 1$
$x \geq 0$
None of these

The correct answer is: $x greater or equal than 0$

Let $f open parentheses x close parentheses equals 1 plus x log subscript e end subscript invisible function application left parenthesis x plus square root of x to the power of 2 end exponent plus 1 end root right parenthesis minus square root of 1 plus x to the power of 2 end exponent end root$
Clearly, $f left parenthesis x right parenthesis$ is defined for all $x element of R$

Now,

$f to the power of ´ end exponent open parentheses x close parentheses equals log subscript e end subscript invisible function application open parentheses x plus square root of x to the power of 2 end exponent plus 1 end root close parentheses plus fraction numerator x over denominator square root of x to the power of 2 end exponent plus 1 end root end fraction minus fraction numerator x over denominator square root of x to the power of 2 end exponent plus 1 end root end fraction$

$rightwards double arrow f to the power of ´ end exponent open parentheses x close parentheses equals log subscript e end subscript invisible function application open parentheses x plus square root of x to the power of 2 end exponent plus 1 end root close parentheses$

$rightwards double arrow f to the power of ´ end exponent open parentheses x close parentheses greater than 0$ for all $x element of R left square bracket because x plus square root of x to the power of 2 end exponent plus 1 end root greater or equal than 1 blank f o r blank a l l blank x element of R right square bracket$

$rightwards double arrow f left parenthesis x right parenthesis$ is increasing on $R$

$rightwards double arrow f open parentheses x close parentheses greater or equal than f open parentheses 0 close parentheses blank f o r blank a l l blank x greater or equal than 0$

$rightwards double arrow 1 plus x log subscript e end subscript invisible function application open parentheses x plus square root of x to the power of 2 end exponent plus 1 end root close parentheses minus square root of 1 plus x to the power of 2 end exponent end root greater or equal than 0 blank f o r blank a l l blank x greater or equal than 0$

$rightwards double arrow 1 plus x log subscript e end subscript invisible function application open parentheses x plus square root of x to the power of 2 end exponent plus 1 end root close parentheses greater or equal than square root of 1 plus x to the power of 2 end exponent end root f o r blank a l l blank x greater or equal than 0$

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