How do you find the critical numbers for y = x/(x^2 + 25) to determine the maximum and minimum?

Jul 7, 2017

By definition the critical numbers of a function are the values of $x$ for which:

$f ' \left(x\right) = 0$

For $f \left(x\right) = \frac{x}{{x}^{2} + 25}$ we have that:

$f ' \left(x\right) = \frac{\left({x}^{2} + 25\right) \frac{d}{\mathrm{dx}} x - x \left(\frac{d}{\mathrm{dx}} \left({x}^{2} + 25\right)\right)}{{x}^{2} + 25} ^ 2 = \frac{{x}^{2} + 25 - 2 {x}^{2}}{{x}^{2} + 25} ^ 2$

$f ' \left(x\right) = - \frac{{x}^{2} - 25}{{x}^{2} + 25} ^ 2$

As the denominator is always positive:

$f ' \left(x\right) = 0 \implies \left({x}^{2} - 25\right) = 0$

So the critical points are $x = \pm 5$ and we can see that:

$f ' \left(x\right) < 0$ for $\left\mid x \right\mid > 5$ and

$f ' \left(x\right) > 0$ for $\left\mid x \right\mid < 5$

which means that $f \left(x\right)$ is decreasing in the intervals $\left(- \infty , - 5\right)$ and $\left(5 , + \infty\right)$ and increasing in the interval $\left(- 5 , 5\right)$.

Thus $x = - 5$ is a local minimum and $x = 5$ is a local maximum.

graph{x/(x^2+25) [-10, 10, -0.15, 0.15]}