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

2 Answers
May 11, 2018

Answer:

At #x = -2#, we have a global maximum of #infty# and a global minimum of #-infty#. A local minimum occurs at #x = 4#.

Explanation:

Critical numbers are the #x# values for which #f'(x) = 0#. These critical points may or may not be maximums/minimums. We first need to find the derivative. We do so using the quotient rule.

#f(x) = (2-x)/(x+2)^3#
#f'(x) = ( ((x+2)^3)(-1) - (2-x)(3)(x+2)^2)/(x+2)^6#
#f'(x) = ( -(x+2)^3 - 3(2-x)(x+2)^2 ) / (x+2)^6#

Set this equal to zero and find which values of #x# satisfy the equation.

#(-(x+2)^3 - 3(2-x)(x+2)^2)/(x+2)^6 = 0#
#-(x+2)^3 - 3(2-x)(x+2)^2 = 0#
#-(x+2)^3 = 3(2-x)(x+2)^2#
#-(x+2) = 3(2-x)#
#-x - 2 = 6 - 3x#
#2x = 8#
#x = 4#

Note that #x = -2# is also a critical number, since the denominator of #f'(x)# will be #0# when #x = -2#. Investigating this critical number, we see that as #x -> -2# from the right, the denominator of #f(x)# gets very small and both the numerator and the denominator remain positive. This implies that the limit of #f(x)# as #x -> -2# from the right explodes to positive infinity. Similarly, as #x -> -2# from the left, #f(x)# explodes to negative infinity. Thus, the global maximum and minimum of the function are #infty# and #-infty#, respectively.

Now let's investigate the critical number #x = 4#. We see that:

#f'(0) = (-8 - 3(2)(8))/(64) = -56/(64)#.

Also,

#f'(5) = (-7^3 - 3(-3)(7^2))/(7^6) = (-343 + 441)/(7^6) = 98/7^6#.

This tells us that before #x = 4#, the function is decreasing and after #x = 4#, the function is increasing. Thus, at #x = 4#, the function must have a local minimum.

I have solved this way. Please, see the answer below:
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