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Maths-

General

Easy

Question

The function $f open parentheses x close parentheses equals log subscript e end subscript invisible function application open parentheses x to the power of 3 end exponent plus square root of x to the power of 6 end exponent plus 1 end root close parentheses$ is of the following types:

Even and increasing
Odd and increasing
Even and decreasing
Odd and decreasing

The correct answer is: Odd and increasing

We have,
$f open parentheses x close parentheses plus f open parentheses negative x close parentheses equals log invisible function application 1 equals 0 rightwards double arrow f left parenthesis negative x right parenthesis equals negative f left parenthesis x right parenthesis$

So, $f left parenthesis x right parenthesis$ is an odd function

Now,

$f open parentheses x close parentheses equals log invisible function application open parentheses x to the power of 3 end exponent plus square root of x to the power of 6 end exponent plus 1 end root close parentheses$

$rightwards double arrow f to the power of ´ end exponent open parentheses x close parentheses equals fraction numerator 1 over denominator x to the power of 3 end exponent plus square root of x to the power of 6 end exponent plus 1 end root end fraction open curly brackets 3 x to the power of 2 end exponent plus fraction numerator 6 x to the power of 5 end exponent over denominator 2 square root of x to the power of 6 end exponent plus 1 end root end fraction close curly brackets equals fraction numerator 3 x to the power of 2 end exponent over denominator square root of x to the power of 6 end exponent plus 1 end root end fraction greater than 0$

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

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