Is #f(x)=(-3x^2-3x-2)/(x^2+x)# increasing or decreasing at #x=1#?

1 Answer
Jim G.
Jun 23, 2017

#"increasing at x = 1"#

Explanation:

#"to determine if f(x) is increasing/decreasing at x = a"#

#• " if " f'(a)>0" then f(x) increasing at x = a"#

#• " if " f'(a)<0" then f(x) decreasing at x = a"#

#"differentiate f(x) using the "color(blue)"quotient rule"#

#"given " f(x)=(g(x))/(h(x))" then"#

#f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2#

#g(x)=-3x^2-3x-2rArrg'(x)=-6x-3#

#h(x)=x^2+xrArrh'(x)=2x+1#

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

#rarrf'(1)=(2.(-9)-(-8).3)/4#

#color(white)(rArrf'(1))=6/4=3/2#

#"since " f'(1)>0" then f(x) is increasing at x = 1"#

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