Is #f(x)=(x^3-4x^2-4x+5)/(x+2)# increasing or decreasing at #x=3#?
1 Answer
Jul 27, 2017
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 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))^2larr" quotient rule"#
#g(x)=x^3-4x^2-4x+5rArrg'(x)=3x^2-8x-4#
#h(x)=x+2rArrh'(x)=1#
#rArrf'(x)=((x+2)(3x^2-8x-4)-(x^3-4x^2-4x+5))/(x+2)^2#
#rArrf'(3)=(5(-1)-(-16))/25=11/25#
#f'(3)>0" hence f(x) is increasing at x = 3"#
graph{(x^3-4x^2-4x+5)/(x+2) [-10, 10, -5, 5]}
