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what is the minim and maxim values f or 1+2cos 4x  for all real numbers ?

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greenbel | Honors

Posted July 4, 2013 at 4:56 PM via web

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what is the minim and maxim values f or 1+2cos 4x  for all real numbers ?

Tagged with math, maxim, minim, value

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted July 7, 2013 at 6:16 PM (Answer #3)

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Differentiating the given function `y = 1 + 2cos 4x` yields:

`(dy)/(dx) = 0 - 8*sin(4x) => (dy)/(dx) = -8*sin(4x)`

You need to solve for x the equation `(dy)/(dx) = 0` to evaluate the critical values of the given function, such that:

`-8*sin(4x) = 0 => sin(4x) = 0 => 4x = (-1)^n sin^(-1) 0 + n*pi`

`4x = 0 + npi => 4x = npi => x = (npi)/4 => {(x = 0),(x = p+-i/4)}`

Replacing 0 for x in equation of the function, yields:

`f(0) = 1 + 2cos4*(0) => f(0) = 1 + 2 = 3`

Replacing `pi/4` for x in equation of the function, yields:

`f(pi/4) = 1 + 2cos4*(pi/4) => f(pi/4) = 1 + 2cos pi = 1 - 2 = -1`

Hence, since the sine function increases over `[0,pi/2]` yields that the function has a minimum value `y = -1` at `x = pi/4` and a maximum value `y = 3,` at `x = 0.`

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jeew-m | College Teacher | (Level 1) Educator Emeritus

Posted July 4, 2013 at 5:08 PM (Answer #1)

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We know that cosine is always between -1 and +1.
Maximum value of a cosine is +1 and minimum is -1.

So when we consider cos4x maximum would be +1.

And when we consider cos4x minimum would be -1.

`f(x) = 1+2cos4x`

Maximum of `f(x) = 1+2xx1 = 3`

Minimum of `f(x) = 1+2xx(-1) = -1`

Sources:

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aruv | High School Teacher | Valedictorian

Posted July 4, 2013 at 5:35 PM (Answer #2)

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`f(x)=1+2cos(4x)`

`f'(x)=1-8sin(4x)`

`` For maximumor minimum ,f'(x)=0

1-8sin(4x)=0

8sin(4x)=1

sin(4x)=1/8

Solving this equation will give the point of maxima/minima

But if we estimate max /min

`-1<=cos(4x)<=1`

`-2<=cos(4x)<=2`

`1-2<=1+cos(4x)<=1+3`

`max f=3`

`and `

`minf=-1`

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