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

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Answer 1 · 2017-07-07T23:24:48

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

$f'(x) = 0$

For $f(x) = x/(x^2+25)$ we have that:

$f'(x) = ((x^2+25)d/dx x- x(d/dx (x^2+25)))/(x^2+25)^2 = (x^2+25-2x^2)/(x^2+25)^2$

$f'(x) = -(x^2-25)/(x^2+25)^2$

As the denominator is always positive:

$f'(x) = 0 => (x^2-25) = 0$

So the critical points are $x=+-5$ and we can see that:

$f'(x) < 0$ for $abs x > 5$ and

$f'(x) > 0$ for $abs x < 5$

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

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]}

How do you find the critical numbers for y = x/(x^2 + 25)y=xx2+25y = x/(x^2 + 25) to determine the maximum and minimum?