How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for f(x)= 2x+3/x?

1 Answer
Andrea S.
Dec 10, 2017

Evaluate the first derivative of the function:

f'(x) = d/dx (2x+3/x) = 2-3/x^2 = (2x^2-3)/x^2

as the denominator is always positive, the sign of f'(x) is the sign of the numerator, which means that f'(x) < 0 when:

2x^2-3 <0

absx < sqrt(3/2)

So the function f(x) is increasing in (-oo, -sqrt(3/2)), decreasing in (-sqrt(3/2), sqrt(3/2)) and again increasing in (sqrt(3/2),+oo). For x=-sqrt(3/2) the function has a local maximum and for x=sqrt(3/2) it has a local minimum.

graph{2x+3/x [-20, 20, -10, 10]}

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