What is the extreme f(t)=2cos(t)+sin(2t) ?

The extreme f(t)=2cos(t)+sin(2t) is

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extreme f(t)=2cos(t)+sin(2t)

extreme points

f (t) = 2 cos (t) + sin (2 t)

Maximum (\frac{π}{6} + 2 πn, 3 + \frac{3}{2}), Minimum (\frac{5 π}{6} + 2 πn, - 3 - \frac{3}{2}), Saddle (\frac{3 π}{2} + 2 πn, 0)

Solution steps

Find the critical points:

t = \frac{π}{6} + 2 πn, t = \frac{5 π}{6} + 2 πn, t = \frac{3 π}{2} + 2 πn

Domain of

2 cos (t) + sin (2 t) : - \infty < t < \infty

Find intervals:

Increasing

: 2 πn \leq t < \frac{π}{6} + 2 πn,

Decreasing

: \frac{π}{6} + 2 πn < t < \frac{5 π}{6} + 2 πn,

Increasing

: \frac{5 π}{6} + 2 πn < t < \frac{3 π}{2} + 2 πn,

Increasing

: \frac{3 π}{2} + 2 πn < t < 2 π + 2 πn

Plug the extreme point

x = \frac{π}{6} + 2 πn

into

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

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

Plug the extreme point

x = \frac{5 π}{6} + 2 πn

into

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

Minimum (\frac{5 π}{6} + 2 πn, - 3 - \frac{3}{2})

Plug the extreme point

x = \frac{3 π}{2} + 2 πn

into

2 cos (t) + sin (2 t) \Rightarrow y = 0

Saddle (\frac{3 π}{2} + 2 πn, 0)

Maximum (\frac{π}{6} + 2 πn, 3 + \frac{3}{2}), Minimum (\frac{5 π}{6} + 2 πn, - 3 - \frac{3}{2}), Saddle (\frac{3 π}{2} + 2 πn, 0)

f (x) = x^{2} - 4

f (x) = 7 x^{2} + \frac{3}{x}

w (x, y) = x + y

f (x, y) = 3 x - 2 y + 6 x y

f (x) = - 2 x^{3} + 6 x - 1