What is the extreme f(x)=5x+10sin(x) ?

The extreme f(x)=5x+10sin(x) is

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extreme f(x)=5x+10sin(x)

extreme

Maximum (\frac{2 π}{3} + 2 πn, 5 (\frac{2 π}{3} + 2 πn) + 10 sin (\frac{2 π}{3} + 2 πn)), Minimum (\frac{4 π}{3} + 2 πn, 5 (\frac{4 π}{3} + 2 πn) + 10 sin (\frac{4 π}{3} + 2 πn))

Solution steps

Find the critical points:

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

Domain of

5 x + 10 sin (x) : - \infty < x < \infty

Find intervals:

Increasing

: 2 πn \leq x < \frac{2 π}{3} + 2 πn,

Decreasing

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

Increasing

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

Plug the extreme point

x = \frac{2 π}{3} + 2 πn

into

5 x + 10 sin (x) \Rightarrow y = 5 (\frac{2 π}{3} + 2 πn) + 10 sin (\frac{2 π}{3} + 2 πn)

Maximum (\frac{2 π}{3} + 2 πn, 5 (\frac{2 π}{3} + 2 πn) + 10 sin (\frac{2 π}{3} + 2 πn))

Plug the extreme point

x = \frac{4 π}{3} + 2 πn

into

5 x + 10 sin (x) \Rightarrow y = 5 (\frac{4 π}{3} + 2 πn) + 10 sin (\frac{4 π}{3} + 2 πn)

Minimum (\frac{4 π}{3} + 2 πn, 5 (\frac{4 π}{3} + 2 πn) + 10 sin (\frac{4 π}{3} + 2 πn))

Maximum (\frac{2 π}{3} + 2 πn, 5 (\frac{2 π}{3} + 2 πn) + 10 sin (\frac{2 π}{3} + 2 πn)), Minimum (\frac{4 π}{3} + 2 πn, 5 (\frac{4 π}{3} + 2 πn) + 10 sin (\frac{4 π}{3} + 2 πn))