What is the extreme f(x)=6sin(x)+cot(x) ?

The extreme f(x)=6sin(x)+cot(x) is

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extreme f(x)=6sin(x)+cot(x)

extreme points

f (x) = 6 sin (x) + cot (x)

Minimum (0.44406 \dots + 2 πn, 4.67960 \dots), Maximum (1.39820 \dots + 2 πn, 6.08518 \dots), Minimum (4.88497 \dots + 2 πn, - 6.08518 \dots), Maximum (5.83912 \dots + 2 πn, - 4.67960 \dots)

Solution steps

Find the critical points:

x = 0.44406 \dots + 2 πn, x = 1.39820 \dots + 2 πn, x = 4.88497 \dots + 2 πn, x = 5.83912 \dots + 2 πn

Domain of

6 sin (x) + cot (x) : 2 πn < x < π + 2 πn or π + 2 πn < x < 2 π + 2 πn

Find intervals:

Decreasing

: 2 πn < x < 0.44406 \dots + 2 πn,

Increasing

: 0.44406 \dots + 2 πn < x < 1.39820 \dots + 2 πn,

Decreasing

: 1.39820 \dots + 2 πn < x < π + 2 πn,

Decreasing

: π + 2 πn < x < 4.88497 \dots + 2 πn,

Increasing

: 4.88497 \dots + 2 πn < x < 5.83912 \dots + 2 πn,

Decreasing

: 5.83912 \dots + 2 πn < x < 2 π + 2 πn

Plug the extreme point

x = 0.44406 \dots + 2 πn

into

6 sin (x) + cot (x) \Rightarrow y = 4.67960 \dots

Minimum (0.44406 \dots + 2 πn, 4.67960 \dots)

Plug the extreme point

x = 1.39820 \dots + 2 πn

into

6 sin (x) + cot (x) \Rightarrow y = 6.08518 \dots

Maximum (1.39820 \dots + 2 πn, 6.08518 \dots)

Plug the extreme point

x = 4.88497 \dots + 2 πn

into

6 sin (x) + cot (x) \Rightarrow y = - 6.08518 \dots

Minimum (4.88497 \dots + 2 πn, - 6.08518 \dots)

Plug the extreme point

x = 5.83912 \dots + 2 πn

into

6 sin (x) + cot (x) \Rightarrow y = - 4.67960 \dots

Maximum (5.83912 \dots + 2 πn, - 4.67960 \dots)

Minimum (0.44406 \dots + 2 πn, 4.67960 \dots), Maximum (1.39820 \dots + 2 πn, 6.08518 \dots), Minimum (4.88497 \dots + 2 πn, - 6.08518 \dots), Maximum (5.83912 \dots + 2 πn, - 4.67960 \dots)

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