What is the extreme (cos(x))/(cos(2x)) ?

The extreme (cos(x))/(cos(2x)) is Minimum(2pin,1),Maximum(pi+2pin,-1)

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extreme (cos(x))/(cos(2x))

extreme points

\frac{cos ( x )}{cos ( 2 x )}

Minimum (2 πn, 1), Maximum (π + 2 πn, - 1)

Solution steps

Find the critical points:

x = 2 πn, x = π + 2 πn

Domain of

\frac{cos ( x )}{cos ( 2 x )} : 2 πn \leq x < \frac{π}{4} + 2 πn or \frac{π}{4} + 2 πn < x < \frac{3 π}{4} + 2 πn or \frac{3 π}{4} + 2 πn < x < \frac{5 π}{4} + 2 πn or \frac{5 π}{4} + 2 πn < x < \frac{7 π}{4} + 2 πn or \frac{7 π}{4} + 2 πn < x < 2 π + 2 πn

Find intervals:

Increasing

: 2 πn < x < \frac{π}{4} + 2 πn,

Increasing

: \frac{π}{4} + 2 πn < x < \frac{3 π}{4} + 2 πn,

Increasing

: \frac{3 π}{4} + 2 πn < x < π + 2 πn,

Decreasing

: π + 2 πn < x < \frac{5 π}{4} + 2 πn,

Decreasing

: \frac{5 π}{4} + 2 πn < x < \frac{7 π}{4} + 2 πn,

Decreasing

: \frac{7 π}{4} + 2 πn < x < 2 π + 2 πn

Plug the extreme point

x = 2 πn

into

\frac{cos ( x )}{cos ( 2 x )} \Rightarrow y = 1

Minimum (2 πn, 1)

Plug the extreme point

x = π + 2 πn

into

\frac{cos ( x )}{cos ( 2 x )} \Rightarrow y = - 1

Maximum (π + 2 πn, - 1)

Minimum (2 πn, 1), Maximum (π + 2 πn, - 1)

f (x) = (\frac{e ^{x}}{5 + e ^{x}})

f (x, y) = - 2 x^{2} + 2 x y - 2 x

\frac{180 ln ( x ) - 300}{x ^{2}}

f (x, y) = x^{2} + y^{2} - 4

f (x) = a + 2 b = 53

What is the extreme (cos(x))/(cos(2x)) ?
The extreme (cos(x))/(cos(2x)) is Minimum(2pin,1),Maximum(pi+2pin,-1)