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$f open parentheses x close parentheses equals 4 tan invisible function application x minus tan to the power of 2 end exponent invisible function application x plus tan to the power of 3 end exponent invisible function application x comma blank x not equal to n pi plus fraction numerator pi over denominator 2 end fraction$

Is monotonically increasing
Is monotonically decreasing
Has a point of maxima
Has a point of minima

The correct answer is: Is monotonically increasing

Here, $f open parentheses x close parentheses equals 4 tan invisible function application x minus tan to the power of 2 end exponent invisible function application x plus tan to the power of 3 end exponent invisible function application x$
$rightwards double arrow f to the power of ´ end exponent open parentheses x close parentheses equals 4 sec to the power of 2 end exponent invisible function application x minus 2 tan invisible function application x sec to the power of 2 end exponent invisible function application x plus 3 tan to the power of 2 end exponent invisible function application x sec to the power of 2 end exponent invisible function application x$
$equals sec to the power of 2 end exponent invisible function application x left parenthesis 4 minus 2 tan invisible function application x plus 3 tan to the power of 2 end exponent invisible function application x right parenthesis$
$equals 3 sec to the power of 2 end exponent invisible function application x open curly brackets tan to the power of 2 end exponent invisible function application x minus fraction numerator 2 over denominator 3 end fraction tan invisible function application x plus fraction numerator 4 over denominator 3 end fraction close curly brackets$
$equals 3 sec to the power of 2 end exponent invisible function application x open curly brackets open parentheses tan invisible function application x minus fraction numerator 1 over denominator 3 end fraction close parentheses to the power of 2 end exponent plus open parentheses fraction numerator 4 over denominator 3 end fraction minus fraction numerator 1 over denominator 9 end fraction close parentheses close curly brackets$
$equals 3 sec to the power of 2 end exponent invisible function application x open curly brackets open parentheses tan invisible function application x minus fraction numerator 1 over denominator 3 end fraction close parentheses to the power of 2 end exponent plus fraction numerator 11 over denominator 9 end fraction close curly brackets greater than 0 comma blank for all x$
Therefore, $f left parenthesis x right parenthesis$ is increasing for all $x element of$ domain

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$increment subscript 1 end subscript equals open vertical bar table row cell y to the power of 5 end exponent z to the power of 6 end exponent left parenthesis z to the power of 3 end exponent minus y to the power of 3 end exponent right parenthesis end cell cell x to the power of 4 end exponent z to the power of 6 end exponent left parenthesis x to the power of 3 end exponent minus z to the power of 3 end exponent right parenthesis end cell cell x to the power of 4 end exponent y to the power of 5 end exponent left parenthesis y to the power of 3 end exponent minus x to the power of 3 end exponent right parenthesis end cell row cell y to the power of 2 end exponent z to the power of 3 end exponent left parenthesis y to the power of 6 end exponent minus z to the power of 6 end exponent right parenthesis end cell cell x z to the power of 3 end exponent left parenthesis z to the power of 6 end exponent minus x to the power of 6 end exponent right parenthesis end cell cell x y to the power of 2 end exponent left parenthesis x to the power of 6 end exponent minus y to the power of 6 end exponent right parenthesis end cell row cell y to the power of 2 end exponent z to the power of 3 end exponent left parenthesis z to the power of 3 end exponent minus y to the power of 3 end exponent right parenthesis end cell cell x z to the power of 3 end exponent left parenthesis x to the power of 3 end exponent minus z to the power of 3 end exponent right parenthesis end cell cell x y to the power of 2 end exponent left parenthesis y to the power of 3 end exponent minus x to the power of 3 end exponent right parenthesis end cell end table close vertical bar$ and $increment subscript 2 end subscript equals open vertical bar table row x cell y to the power of 2 end exponent end cell cell z to the power of 3 end exponent end cell row cell x to the power of 4 end exponent end cell cell y to the power of 5 end exponent end cell cell z to the power of 6 end exponent end cell row cell x to the power of 7 end exponent end cell cell y to the power of 8 end exponent end cell cell z to the power of 9 end exponent end cell end table close vertical bar$ . Then $increment subscript 1 end subscript increment subscript 2 end subscript$ is equal to

The system of equations $a x minus y minus z equals alpha minus 1$ $x minus alpha y minus z equals alpha minus 1$ $x minus y minus alpha z equals alpha minus 1$ Has no solution if $alpha$ is

The system of equations $a x minus y minus z equals alpha minus 1$ $x minus alpha y minus z equals alpha minus 1$ $x minus y minus alpha z equals alpha minus 1$ Has no solution if $alpha$ is

The value of the determinant $open vertical bar table row 1 1 1 row cell blank to the power of m end exponent C subscript 1 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 1 end subscript end cell cell blank to the power of m plus 2 end exponent C subscript 1 end subscript end cell row cell blank to the power of m end exponent C subscript 2 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 2 end subscript end cell cell blank to the power of m plus 2 end exponent C subscript 2 end subscript end cell end table close vertical bar$ is equal to

The value of the determinant $open vertical bar table row 1 1 1 row cell blank to the power of m end exponent C subscript 1 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 1 end subscript end cell cell blank to the power of m plus 2 end exponent C subscript 1 end subscript end cell row cell blank to the power of m end exponent C subscript 2 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 2 end subscript end cell cell blank to the power of m plus 2 end exponent C subscript 2 end subscript end cell end table close vertical bar$ is equal to

parallel

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