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Popular >

extreme f(x)=2x+sin(4x)

Solution

extreme points

f (x) = 2 x + sin (4 x)

Solution

Maximum (\frac{π}{6} + \frac{πn}{2}, \frac{π}{3} + πn + sin (4 (\frac{π}{6} + \frac{πn}{2}))), Minimum (\frac{π}{3} + \frac{πn}{2}, \frac{2 π}{3} + πn + sin (4 (\frac{π}{3} + \frac{πn}{2})))

Solution steps

Find the critical points:

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

Domain of

2 x + sin (4 x) : - \infty < x < \infty

Find intervals:

Increasing

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

Decreasing

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

Increasing

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

Plug the extreme point

x = \frac{π}{6} + \frac{πn}{2}

into

2 x + sin (4 x) \Rightarrow y = \frac{π}{3} + πn + sin (4 (\frac{π}{6} + \frac{πn}{2}))

Maximum (\frac{π}{6} + \frac{πn}{2}, \frac{π}{3} + πn + sin (4 (\frac{π}{6} + \frac{πn}{2})))

Plug the extreme point

x = \frac{π}{3} + \frac{πn}{2}

into

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

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

Maximum (\frac{π}{6} + \frac{πn}{2}, \frac{π}{3} + πn + sin (4 (\frac{π}{6} + \frac{πn}{2}))), Minimum (\frac{π}{3} + \frac{πn}{2}, \frac{2 π}{3} + πn + sin (4 (\frac{π}{3} + \frac{πn}{2})))

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extreme f(x)=ye^{-x}extreme points

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Frequently Asked Questions (FAQ)

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