Question

Please, help me solve it! Or give me a method!?

Answer 1

See below.

Explanation:

Taking a)

If we have

`f(x)g(x)=0 rArr {(f(x)=0),(g(x)=0):}`

Calling

`f(x) = Cos(2 x) - 3 Cos(x) - 1 = 2cos^2(x)-3cos(x)-2`

and

`g(x) = sqrt(log_(1/3)(x-2)+2)`

then

`f(x)=0 rArr 2cos^2(x)-3cos(x)-2` or

`x = {pm 2pi/3 +2kpi} uu {pm "arccos"(2)+2kpi}`

Now

`sqrt(log_(1/3)(x-2)+2)=0 rArr log_(1/3)(x-2)+2=0 rArr log_(1/3)(x-2)=-2 rArr(1/3)^(-2)=x-2 rArr x = 2+(1/3)^(-2)`

and finally

`f(x)g(x) = 0` occurs for

`x = {pm 2pi/3 +2kpi} uu {pm "arccos"(2)+2kpi} uu {x = 2+(1/3)^(-2)}`

Part b) is left as an exercise.