# Is #f(x)=(-5x^3-x^2-3x-11)/(x-3)# increasing or decreasing at #x=2#?

##### 1 Answer

Mar 6, 2017

#### Explanation:

To determine if f(x) is increasing /decreasing at x = a, evaluate f'(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"#

#color(red)(bar(ul(|color(white)(2/2)color(black)(f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2)color(white)(2/2)|)))#

#"here "g(x)=-5x^3-x^2-3x-11#

#rArrg'(x)=-15x^2-2x-3#

#"and "h(x)=x-3rArrh'(x)=1#

#rArrf'(x)=((x-3)(-15x^2-2x-3)-(-5x^3-x^2-3x-11)(1))/(x-3)^2#

#rArrf'(2)=((-1)(-67)-(-61))/1=128# Since f'(2) > 0, then f(x) is increasing at x = 2

graph{(-5x^3-x^2-3x-11)/(x-3) [-10, 10, -5, 5]}