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

1 Answer
Andrea S.
Dec 4, 2016

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]}

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