Intermediate value theorem.
Extrema occur at points where
![\dfrac{\mathrm dy}{\mathrm dx}=0](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D%3D0)
, with maxima occurring at
![x=c](https://tex.z-dn.net/?f=x%3Dc)
if the derivative is positive to the left of
![c](https://tex.z-dn.net/?f=c)
and negative to the right of
![c](https://tex.z-dn.net/?f=c)
, and minima in the opposite case.
So suppose you take two values
![a,b](https://tex.z-dn.net/?f=a%2Cb)
. If it turns out that
![\dfrac{\mathrm dy}{\mathrm dx}\bigg|_{x=a}>0](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D%5Cbigg%7C_%7Bx%3Da%7D%3E0)
and
![\dfrac{\mathrm dy}{\mathrm dx}\bigg|_{x=b}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D%5Cbigg%7C_%7Bx%3Db%7D%3C0)
, then the IVT guarantees the existence of some
![c\in(a,b)](https://tex.z-dn.net/?f=c%5Cin%28a%2Cb%29)
such that
![\dfrac{\mathrm dy}{\mathrm dx}\bigg|_{x=c}=0](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D%5Cbigg%7C_%7Bx%3Dc%7D%3D0)
.
Choosing arbitrary values of
![a,b](https://tex.z-dn.net/?f=a%2Cb)
won't guarantee that exactly one such
![c](https://tex.z-dn.net/?f=c)
exists, though. The function could easily oscillate several more times between
![a](https://tex.z-dn.net/?f=a)
and
![b](https://tex.z-dn.net/?f=b)
, intersecting the x-axis more than once, for example. This is where your suspicion can be applied. Knowing that
![\cos x=0](https://tex.z-dn.net/?f=%5Ccos%20x%3D0)
for
![x=\dfrac\pi2,\dfrac{3\pi}2,\dfrac{7\pi}2](https://tex.z-dn.net/?f=x%3D%5Cdfrac%5Cpi2%2C%5Cdfrac%7B3%5Cpi%7D2%2C%5Cdfrac%7B7%5Cpi%7D2)
(approximately 1.57, 4.71, 7.85, respectively), you can use these values as reference points for computing the sign of the derivative.
When
![x=\dfrac\pi2](https://tex.z-dn.net/?f=x%3D%5Cdfrac%5Cpi2)
, you have
![\dfrac{\mathrm dy}{\mathrm dx}=-\dfrac18](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D%3D-%5Cdfrac18)
. You know that
![\cos x>0](https://tex.z-dn.net/?f=%5Ccos%20x%3E0)
for
![0](https://tex.z-dn.net/?f=0%3Cx%3C%5Cdfrac%5Cpi2)
, and that as
![x\to0^+](https://tex.z-dn.net/?f=x%5Cto0%5E%2B)
,
![\dfrac{\mathrm dy}{\mathrm dx}\to+\infty](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D%5Cto%2B%5Cinfty)
. This means there must be some
![c\in\left(0,\dfrac\pi2\right)](https://tex.z-dn.net/?f=c%5Cin%5Cleft%280%2C%5Cdfrac%5Cpi2%5Cright%29)
such that
![\dfrac{\mathrm dy}{\mathrm dx}=0](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D%3D0)
, and in particular, this value of
![c](https://tex.z-dn.net/?f=c)
is the site of a relative maximum.
You can use similar arguments to determine what happens at the other two suspected critical points.