Is #f(x)=(6x^2-x-12)/(x+3)# increasing or decreasing at #x=3#?

1 Answer
Jul 10, 2017

#"increasing at " x=3#

Explanation:

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

#"differentiate and evaluate at x = a"#

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

#• " if " f'(a)<0" then f(x) is 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)=6x^2-x-12rArrg'(x)=12x-1#

#h(x)=x+3rArrh'(x)=1#

#rArrf'(x)=((x+3)(12x-1)-(6x^2-x-12))/(x+3)^2#

#rArrf'(3)=(210-39)/36>0#

#rArrf(x)" is increasing at "x=3#