Question

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

Answer 1

By definition the critical numbers of a function are the values of `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]}