For f(x)= 6x+(8/x) where is the function increasing and decreasing and where are the maximum and minimum points.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The function f(x) = `6x + 8/x`

f'(x) = `6 - 8/x^2`

f'(x) > 0

=> `6 - 8/x^2 > 0`

=> `6x^2 > 8`

=> `x^2 > 4/3`

=> x lies in `(-oo, -2/sqrt 3)U(2/sqrt 3 , oo)`

f'(x) < 0

=> `x^2 < 4/3`

=> x lies in `(-2/sqrt 3, 2/sqrt 3)`

f'(x) = 0

=> x^2 = 4/3

=> x = `-2/sqrt 3` and x = `2/sqrt 3`

f''(x) = `8/x^3`

This is negative for x = `-2/sqrt 3` indicating a maximum point at x = `-2/sqrt 3` and as f''(x) is positive for x =` 2/sqrt 3` there is a minimum point at x = `2/sqrt 3` .

The function is increasing in `(-oo, -2/sqrt 3)U(2/sqrt 3 , oo)` , it is decreasing in `(-2/sqrt 3, 2/sqrt 3)` , the maximum is at x = `-2/sqrt 3` and the minimum at x = `2/sqrt 3`

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