# How do you find the critical numbers of f(x)=sinxcosx?

Feb 1, 2017

Critical numbers of $f \left(x\right)$ occur when

$x \setminus \setminus = \frac{\pi}{4} + \frac{n \pi}{2} \setminus \setminus \setminus \setminus n \in \mathbb{Z}$
eg $x = \pm \frac{\pi}{4} , \pm 3 \frac{\pi}{4} , \pm 5 \frac{\pi}{4} , \ldots$

#### Explanation:

We have:

$f \left(x\right) = \sin x \cos x$

Differentiating wrt $x$ using the product rule:

$f ' \left(x\right) = \left(\sin x\right) \left(- \sin x\right) + \left(\cos x\right) \left(\cos x\right)$
$\text{ } = {\cos}^{2} x - {\sin}^{2} x$
$\text{ } = \cos \left(2 x\right)$

At a critical point $f ' \left(x\right) = 0$

$\therefore \cos \left(2 x\right) = 0$
$\therefore 2 x = \frac{\pi}{2} + n \pi$
$\therefore x \setminus \setminus = \frac{\pi}{4} + \frac{n \pi}{2} \setminus \setminus \setminus \setminus n \in \mathbb{Z}$

Hence critical numbers of $f \left(x\right)$ occur when

$x \setminus \setminus = \frac{\pi}{4} + \frac{n \pi}{2} \setminus \setminus \setminus \setminus n \in \mathbb{Z}$
eg $x = \pm \frac{\pi}{4} , \pm 3 \frac{\pi}{4} , \pm 5 \frac{\pi}{4} , \ldots$

We can see these values of $x$ correspond to max/min of the graph $y = \sin x \cos x$: