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Maths-

General

Easy

Question

Which of the following functions is not decreasing on $left parenthesis 0 comma blank pi divided by 2 right parenthesis$ ?

$\cos x$
$\cos 2 x$
$\cos^{2} x$
None of these

The correct answer is: None of these

If $f open parentheses x close parentheses equals open parentheses a b minus b to the power of 2 end exponent minus 2 close parentheses x plus not stretchy integral subscript 0 end subscript superscript x end superscript left parenthesis cos to the power of 4 end exponent invisible function application theta plus sin to the power of 4 end exponent invisible function application theta right parenthesis d theta$ is decreasing function of $x$ for all $x element of R$ and $b element of R comma blank b$ being independent of $x$ , then

$equals a b minus b to the power of 2 end exponent minus 2 plus open parentheses sin to the power of 2 end exponent invisible function application x plus cos to the power of 2 end exponent invisible function application x close parentheses to the power of 2 end exponent minus 2 sin to the power of 2 end exponent invisible function application x cos to the power of 2 end exponent invisible function application x less than 0$

$rightwards double arrow blank a b minus b to the power of 2 end exponent minus 1 less than open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses sin to the power of 2 end exponent invisible function application 2 x less than fraction numerator 1 over denominator 2 end fraction$

$rightwards double arrow blank 2 a b minus 2 b to the power of 2 end exponent minus 2 less than 0$

$rightwards double arrow blank 2 b to the power of 2 end exponent minus 2 a b plus 3 greater than 0$

$therefore open parentheses negative 2 a to the power of 2 end exponent close parentheses minus 4.2.3 less than 0$

$rightwards double arrow a to the power of 2 end exponent less than 6$

$rightwards double arrow blank minus square root of 6 less than a less than square root of 6$

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