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How do use the first derivative test to determine the local extrema #f(x) = x / (x^2+1)#?

1 Answer

Sep 29, 2015

See the explanation.

Explanation:

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

#f'(x)=0 <=> 1-x^2=0 <=> x=1 vv x=-1#

#f'(x)>0# #AAx in (-1,1)# function increasing
#f'(x)<0# #AAx in (-oo,-1)uu(1,oo)# function decreasing

#f'(x)# changes sign at #x=-1# and #f(x)# has a minimum value #f_min=f(-1)=-1/2#

#f'(x)# changes sign at #x=1# and #f(x)# has a maximum value #f_max=f(1)=1/2#

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