# Is f(x)=sinx/x increasing or decreasing at x=pi/3?

Apr 14, 2016

Decreasing.

#### Explanation:

To determine if a function is increasing or decreasing at a point, use the function's derivative:

• If $f ' \left(a\right) < 0$, then $f$ is decreasing at $x = a$.
• If $f ' \left(a\right) > 0$, 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

$f ' \left(x\right) = \frac{x \frac{d}{\mathrm{dx}} \left(\sin x\right) - \sin x \frac{d}{\mathrm{dx}} \left(x\right)}{x} ^ 2$

$= \frac{x \cos x - \sin x}{x} ^ 2$

So, to determine if $f$ is increasing or decreasing at $x = \frac{\pi}{3}$, find $f ' \left(\frac{\pi}{3}\right)$ and see if it is positive or negative.

$f ' \left(\frac{\pi}{3}\right) = \frac{\frac{\pi}{3} \cos \left(\frac{\pi}{3}\right) - \sin \left(\frac{\pi}{3}\right)}{\frac{\pi}{3}} ^ 2$

$= \frac{\frac{\pi}{3} \left(\frac{1}{2}\right) - \frac{\sqrt{3}}{2}}{{\pi}^{2} / 9} \approx - 0.3123$

Since this is $< 0$, the function is decreasing at $x = \frac{\pi}{3}$.

We can check a graph of $f$ (note that pi/3approx1.0472).

graph{sinx/x [-3.945, 4.825, -1.568, 2.817]}