What is the extreme sin(4x) ?

The extreme sin(4x) is Maximum(pi/8+pi/2 n,1),Minimum((3pi)/8+pi/2 n,-1)

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extreme sin(4x)

extreme points

sin (4 x)

Maximum (\frac{π}{8} + \frac{π}{2} n, 1), Minimum (\frac{3 π}{8} + \frac{π}{2} n, - 1)

Solution steps

Find the critical points:

x = \frac{π}{8} + \frac{π}{2} n, x = \frac{3 π}{8} + \frac{π}{2} n

Domain of

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

Find intervals:

Increasing

: \frac{π}{2} n \leq x < \frac{π}{8} + \frac{π}{2} n,

Decreasing

: \frac{π}{8} + \frac{π}{2} n < x < \frac{3 π}{8} + \frac{π}{2} n,

Increasing

: \frac{3 π}{8} + \frac{π}{2} n < x < \frac{π}{2} + \frac{π}{2} n

Plug the extreme point

x = \frac{π}{8} + \frac{π}{2} n

into

sin (4 x) \Rightarrow y = 1

Maximum (\frac{π}{8} + \frac{π}{2} n, 1)

Plug the extreme point

x = \frac{3 π}{8} + \frac{π}{2} n

into

sin (4 x) \Rightarrow y = - 1

Minimum (\frac{3 π}{8} + \frac{π}{2} n, - 1)

Maximum (\frac{π}{8} + \frac{π}{2} n, 1), Minimum (\frac{3 π}{8} + \frac{π}{2} n, - 1)

f (x) = \frac{( x ^{2} - 4 x + 4 )}{x - 8}

f (x) = \frac{x ^{3}}{3} - 10 x^{2} + 75 x + 20

f (x, y) = x^{2} - x y + y^{2} - 3 x + 3 y

\frac{x ^{3} + 1}{x + 1}

What is the extreme sin(4x) ?
The extreme sin(4x) is Maximum(pi/8+pi/2 n,1),Minimum((3pi)/8+pi/2 n,-1)