Trigonomic Function

fgmango

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Oct 13, 2005
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11
The function is f (x) =3sinx+cos2x

f'(x)=3cosx-2sin2x

f"(x)=-3sinx-4cos2x

Is this right?
 
I'm having trouble finding the x-int

f"(x)=-3sinx-fcos2x = -sinx(3+4sinx)

-sinx=0 3+4sinx=o
sinx=0 4sinx=-3
sinx=-3/4

Is this right?
 
What is the "f" in "-3sin(x) - fcos(2x)"?

How are you getting that 4cos(2x) equals 4sin<sup>2</sup>(x)?

Eliz.
 
You might want to check your identities: cos(2x) = 1 + sin<sup>2</sup>(x).

Eliz.
 
Hello, fgmango!

I'm having trouble finding the x-int
No wonder! . . . You're solving: f(x)=0\displaystyle f''(x)\,=\,0
. . You're looking for inflection points!

f(x)=3sin(x)4cos(2x)=0\displaystyle f''(x)\:=\:-3\cdot\sin(x)\,-\,4\cdot\cos(2x)\:=\:0

Then: .-3sin(x)4[12sin2(x)]=0\displaystyle 3\cdot\sin(x)\,-\,4[1\,-\,2\cdot\sin^2(x)]\:=\:0

We have a quadratic: .8sin2(x)3sin(x)4=0\displaystyle 8\cdot\sin^2(x)\,-\,3\cdot\sin(x)\,-\,4\:=\:0

. . with roots: .sin(x)=3±13716\displaystyle \sin(x)\:=\:\frac{3\,\pm\,\sqrt{137}}{16}


Inflection points occur at: .x  =  arcsin(3±13716)\displaystyle x\;=\;\arcsin\left(\frac{3\,\pm\,\sqrt{137}}{16}\right)

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If you're really looking for x\displaystyle x-intercepts, set f(x)=0.\displaystyle f(x)\,=\,0.

We have: .3sin(x)+cos(2x)=0\displaystyle 3\cdot\sin(x)\,+\,\cos(2x)\:=\:0

Then: .3sin(x)+[12sin2(x)]=0\displaystyle 3\cdot\sin(x)\,+\,[1\,-\,2\cdot\sin^2(x)]\:=\:0

We have a quadratic: .2sin2(x)3sin(x)1=0\displaystyle 2\cdot\sin^2(x)\,-\,3\cdot\sin(x)\,-\,1\:=\:0

. . with roots: .sin(x)  =  3±174\displaystyle \sin(x)\;=\;\frac{3\,\pm\,\sqrt{17}}{4}


x\displaystyle x-intercepts occur at: .x=arcsin(3±174)\displaystyle x\:=\:\arcsin\left(\frac{3\,\pm\,\sqrt{17}}{4}\right)
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