What is the extreme f(x)=2cos(x)+sin(2x) ?

The extreme f(x)=2cos(x)+sin(2x) is

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extreme f(x)=2cos(x)+sin(2x)

extreme points

f (x) = 2 cos (x) + sin (2 x)

Maximum (\frac{π}{6} + 2 πn, 3 + \frac{3}{2}), Minimum (\frac{5 π}{6} + 2 πn, - 3 - \frac{3}{2}), Saddle (\frac{3 π}{2} + 2 πn, 0)

Solution steps

Find the critical points:

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{3 π}{2} + 2 πn

Domain of

2 cos (x) + sin (2 x) : - \infty < x < \infty

Find intervals:

Increasing

: 2 πn \leq x < \frac{π}{6} + 2 πn,

Decreasing

: \frac{π}{6} + 2 πn < x < \frac{5 π}{6} + 2 πn,

Increasing

: \frac{5 π}{6} + 2 πn < x < \frac{3 π}{2} + 2 πn,

Increasing

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

Plug the extreme point

x = \frac{π}{6} + 2 πn

into

2 cos (x) + sin (2 x) \Rightarrow y = 3 + \frac{3}{2}

Maximum (\frac{π}{6} + 2 πn, 3 + \frac{3}{2})

Plug the extreme point

x = \frac{5 π}{6} + 2 πn

into

2 cos (x) + sin (2 x) \Rightarrow y = - 3 - \frac{3}{2}

Minimum (\frac{5 π}{6} + 2 πn, - 3 - \frac{3}{2})

Plug the extreme point

x = \frac{3 π}{2} + 2 πn

into

2 cos (x) + sin (2 x) \Rightarrow y = 0

Saddle (\frac{3 π}{2} + 2 πn, 0)

Maximum (\frac{π}{6} + 2 πn, 3 + \frac{3}{2}), Minimum (\frac{5 π}{6} + 2 πn, - 3 - \frac{3}{2}), Saddle (\frac{3 π}{2} + 2 πn, 0)