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Popular >

extreme y=-x+2cos(x)

Solution

extreme points

y = - x + 2 cos (x)

Solution

Minimum (\frac{7 π}{6} + 2 πn, - \frac{7 π}{6} - 2 πn + 2 cos (\frac{7 π}{6} + 2 πn)), Maximum (\frac{11 π}{6} + 2 πn, - \frac{11 π}{6} - 2 πn + 2 cos (\frac{11 π}{6} + 2 πn))

Solution steps

Find the critical points:

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Domain of

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

Find intervals:

Decreasing

: 2 πn \leq x < \frac{7 π}{6} + 2 πn,

Increasing

: \frac{7 π}{6} + 2 πn < x < \frac{11 π}{6} + 2 πn,

Decreasing

: \frac{11 π}{6} + 2 πn < x < 2 π + 2 πn

Plug the extreme point

x = \frac{7 π}{6} + 2 πn

into

- x + 2 cos (x) \Rightarrow y = - \frac{7 π}{6} - 2 πn + 2 cos (\frac{7 π}{6} + 2 πn)

Minimum (\frac{7 π}{6} + 2 πn, - \frac{7 π}{6} - 2 πn + 2 cos (\frac{7 π}{6} + 2 πn))

Plug the extreme point

x = \frac{11 π}{6} + 2 πn

into

- x + 2 cos (x) \Rightarrow y = - \frac{11 π}{6} - 2 πn + 2 cos (\frac{11 π}{6} + 2 πn)

Maximum (\frac{11 π}{6} + 2 πn, - \frac{11 π}{6} - 2 πn + 2 cos (\frac{11 π}{6} + 2 πn))

Minimum (\frac{7 π}{6} + 2 πn, - \frac{7 π}{6} - 2 πn + 2 cos (\frac{7 π}{6} + 2 πn)), Maximum (\frac{11 π}{6} + 2 πn, - \frac{11 π}{6} - 2 πn + 2 cos (\frac{11 π}{6} + 2 πn))

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