Question

How do you find the critical numbers for `h(p) = (p - 2)/(p^2 + 3)` to determine the maximum and minimum?

Answer 1

The critical points are:

`p_1 = 2-sqrt(7)` that is a minimum.
`p_2 = 2+sqrt(7)` that is a maximum.

Explanation:

The critical points of a function are the points for which its derivative is null.

`(dh)/(dp) = frac( (p^2+3) - 2p(p-2) ) ((p^2+3)^2) = -(p^2-4p-3)/((p^2+3)^2)`

The denominator is strictly positive for every `p in RR`, so we can focus on the numerator:

`p^2-4p-3= 0`

`p = 2+- sqrt(4+3) = 2+-sqrt(7)`

To determine whether this point are maximums or minimums we could calculate the second derivative, but as it is easy to see the sign of `(dh)/(dp)` we can proceed this way:

1) Around `p_1=2-sqrt(7)` we have:

`(dh)/(dp) < 0` for `p< p_1`

`(dh)/(dp) > 0` for `p> p_1`

so `h(p)` decreases until `p_1` then increases, so that `2-sqrt(7)` is a minimum.

2) Around `p_2=2+sqrt(7)` we have:

`(dh)/(dp) > 0` for `p< p_2`

`(dh)/(dp) < 0` for `p> p_2`

so `h(p)` increases until `p_2` then decreases, so that `2+sqrt(7)` is a maximum.

graph{(x-2)/(x^2+3) [-10, 10, -1, 1]}