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

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

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Answer 1 · 2017-12-10T16:15:03

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

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

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