Subjective Type

Is $$ f(x)= \dfrac {\sin x} {x} $$ increasing or decreasing at $$ x= \dfrac {\pi} {3} ? $$

Solution

Decreasing.
Explanation:
To determine if a function is increasing or decreasing at a point, use the function's derivative:
If $$ f'(a)<0 ,$$ then $$ f $$ is decreasing at $$ x=a .$$
If $$ f'(a)>0 , $$ then $$ f $$ is increasing at $$ x=a .$$
So, we first must find the derivative of $$ f . $$ To do so, we will have to use the quotient rule. Application of the quotient rule shows that
$$ f'(x)= \dfrac {x \dfrac {d} {dx} (\sin x)−\sin x \dfrac {d} {dx} (x)} {x^2}
\\ = \dfrac {x \cos x−\sin x} {x^2} $$
So, to determine if $$ f $$ is increasing or decreasing at $$ x= \dfrac {\pi} {3} , $$ find $$ f'(\dfrac {\pi} {3} ) $$ and see if it is positive or negative.
$$ f'( \dfrac {\pi} {3} )= \dfrac {\dfrac {\pi} {3} \cos (\dfrac {\pi} {3})−\sin (\dfrac {\pi} {3} )} {(\dfrac {\pi} {3} )^2}
\\ = \dfrac { \dfrac {\pi} {3} (\dfrac {1} {2} )− \dfrac {\sqrt 3} {2} } {\dfrac {\pi^2} {9} } \ approx −0.3123 $$
Since this is $$ <0 ,$$ the function is decreasing at $$ x= \dfrac {\pi} {3} .$$
We can check a graph of f ( note that $$ \dfrac {\pi} {3} \approx 1.0472 $$ ) .
graph{sinx/x [-3.945, 4.825, -1.568, 2.817]}

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