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?

Dec 10, 2017

Evaluate the first derivative of the function:

$f ' \left(x\right) = \frac{d}{\mathrm{dx}} \left(2 x + \frac{3}{x}\right) = 2 - \frac{3}{x} ^ 2 = \frac{2 {x}^{2} - 3}{x} ^ 2$

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

$2 {x}^{2} - 3 < 0$

$\left\mid x \right\mid < \sqrt{\frac{3}{2}}$

So the function $f \left(x\right)$ is increasing in $\left(- \infty , - \sqrt{\frac{3}{2}}\right)$, decreasing in $\left(- \sqrt{\frac{3}{2}} , \sqrt{\frac{3}{2}}\right)$ and again increasing in $\left(\sqrt{\frac{3}{2}} , + \infty\right)$. For $x = - \sqrt{\frac{3}{2}}$ the function has a local maximum and for $x = \sqrt{\frac{3}{2}}$ it has a local minimum.

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