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Popular >

extreme 2θ-4sin(θ)

Solution

extreme points

2 θ - 4 sin (θ)

Solution

Minimum (\frac{π}{3} + 2 πn, 2 (\frac{π}{3} + 2 πn) - 4 sin (\frac{π}{3} + 2 πn)), Maximum (\frac{5 π}{3} + 2 πn, 2 (\frac{5 π}{3} + 2 πn) - 4 sin (\frac{5 π}{3} + 2 πn))

Solution steps

Find the critical points:

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

Domain of

2 θ - 4 sin (θ) : - \infty < θ < \infty

Find intervals:

Decreasing

: 2 πn \leq θ < \frac{π}{3} + 2 πn,

Increasing

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

Decreasing

: \frac{5 π}{3} + 2 πn < θ < 2 π + 2 πn

Plug the extreme point

x = \frac{π}{3} + 2 πn

into

2 θ - 4 sin (θ) \Rightarrow y = 2 (\frac{π}{3} + 2 πn) - 4 sin (\frac{π}{3} + 2 πn)

Minimum (\frac{π}{3} + 2 πn, 2 (\frac{π}{3} + 2 πn) - 4 sin (\frac{π}{3} + 2 πn))

Plug the extreme point

x = \frac{5 π}{3} + 2 πn

into

2 θ - 4 sin (θ) \Rightarrow y = 2 (\frac{5 π}{3} + 2 πn) - 4 sin (\frac{5 π}{3} + 2 πn)

Maximum (\frac{5 π}{3} + 2 πn, 2 (\frac{5 π}{3} + 2 πn) - 4 sin (\frac{5 π}{3} + 2 πn))

Minimum (\frac{π}{3} + 2 πn, 2 (\frac{π}{3} + 2 πn) - 4 sin (\frac{π}{3} + 2 πn)), Maximum (\frac{5 π}{3} + 2 πn, 2 (\frac{5 π}{3} + 2 πn) - 4 sin (\frac{5 π}{3} + 2 πn))

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Frequently Asked Questions (FAQ)

What is the extreme 2θ-4sin(θ) ?
The extreme 2θ-4sin(θ) is