Question #c9e4c

1 Answer
Shura
Jun 6, 2015

f(x) = x/(x-2)

The domain is D = RR-{2}

f'(x) = ((x)'(x-2) - x(x-2)')/(x-2)^2

f'(x) = (x-2-x)/(x-2)^2 = -2/(x-2)^2

To find the local max an min of f(x), you need to study the sign of the derivative :
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You are absolutely right, there aren't any local max or min.

f''(x) = ((-2)'(x-2)^2-(-2)((x-2)^2)')/(x-2)^4

f''(x) = (4(x-2))/(x-2)^4 = 4/(x-2)^3

To find the intervals in which f(x) is concave up or concave down, you need to study the sign of the second derivative :
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The function is concave down for x in ]-oo;+2[ and is concave up for x in ]+2;+oo[.

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