What are the local extrema, if any, of #f (x) =(x^3+3x^2)/(x^2-5x)#?

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Answer 1 · 2018-06-03T12:48:13

#MIN(5+2sqrt(10),4sqrt(10)+13)# and #MAX(5-2*sqrt(10),-4sqrt(10)+13)#

By the Quotient rule we get

#f'(x)=((3x^2+6x)(x^2-5x)-(x^3+3x^2)(2x-5))/(x^2-5x)^2#
simplifying we obtain

#f'(x)=(x^2-10x-15)/(x-5)^2#

#f''(x)=80/(x-5)^3#

so we have

#f''(5+2*sqrt(10))=80/(2sqrt(10))^3>0#

#f''(5-2sqrt(10))=80/(-2sqrt(10))^3<0#