What is the extreme 2cos((2pit)/3) ?

The extreme 2cos((2pit)/3) is Maximum(3n,2),Minimum(3/2+3n,-2)

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extreme 2cos((2pit)/3)

extreme points

2 cos (\frac{2 π t}{3})

Maximum (3 n, 2), Minimum (\frac{3}{2} + 3 n, - 2)

Solution steps

Find the critical points:

t = 3 n, t = \frac{3}{2} + 3 n

Domain of

2 cos (\frac{2 π t}{3}) : - \infty < t < \infty

Find intervals:

Decreasing

: 3 n < t < \frac{3}{2} + 3 n,

Increasing

: \frac{3}{2} + 3 n < t < 3 + 3 n

Plug the extreme point

x = 3 n

into

2 cos (\frac{2 π t}{3}) \Rightarrow y = 2

Maximum (3 n, 2)

Plug the extreme point

x = \frac{3}{2} + 3 n

into

2 cos (\frac{2 π t}{3}) \Rightarrow y = - 2

Minimum (\frac{3}{2} + 3 n, - 2)

Maximum (3 n, 2), Minimum (\frac{3}{2} + 3 n, - 2)

x^{3} - 12 x + 16

x^{3} - 12 x + 12

f (x, y) = x^{2} + 5 x y + y^{2} - 2 x + 4 y + 5

f (x, y) = 9 x y + \frac{1000}{y} + \frac{1000}{x}

f (x) = - 2 x + 6 x^{\frac{2}{3}}, - 8 \leq x \leq 4.5

What is the extreme 2cos((2pit)/3) ?
The extreme 2cos((2pit)/3) is Maximum(3n,2),Minimum(3/2+3n,-2)