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

1 Answer
Dec 4, 2016

Answer:

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]}