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Maths-

General

Easy

Question

The point in the interval $left square bracket 0 comma blank 2 pi right square bracket$ , where $f open parentheses x close parentheses equals e to the power of x end exponent sin invisible function application x$ has maximum slope is

$\frac{π}{4}$
$\frac{π}{2}$
$π$
None of these

The correct answer is: $fraction numerator pi over denominator 2 end fraction$

(Slope) $f to the power of ´ end exponent open parentheses x close parentheses equals e to the power of x end exponent cos invisible function application x plus sin invisible function application x e to the power of x end exponent$
$equals e to the power of x end exponent square root of 2 sin invisible function application left parenthesis x plus pi divided by 4 right parenthesis$

$f to the power of ´ ´ end exponent open parentheses x close parentheses equals square root of 2 e to the power of x end exponent open curly brackets sin invisible function application x plus pi divided by 4 close curly brackets plus cos invisible function application left curly bracket x plus pi divided by 4 right curly bracket$

$equals 2 blank e to the power of x end exponent. sin invisible function application left parenthesis x plus pi divided by 2 right parenthesis$

For maximum slope put, $f to the power of ´ ´ end exponent open parentheses x close parentheses equals 0$

$rightwards double arrow sin invisible function application left parenthesis x plus pi divided by 2 right parenthesis equals 0$

$rightwards double arrow blank cos invisible function application x equals 0$

$therefore blank x equals pi divided by 2 comma blank 3 pi divided by 2$

$f to the power of ´ ´ ´ end exponent open parentheses x close parentheses equals 2 e to the power of x end exponent cos invisible function application left parenthesis x plus pi divided by 2 right parenthesis$

$f to the power of ´ ´ ´ end exponent open parentheses pi divided by 2 close parentheses equals 2 e to the power of x end exponent. cos invisible function application pi equals negative v e$

Maximum slope is at $x equals pi divided by 2$

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