Is #f(x)=(-2x^3+x^2-2x-4)/(4x-2)# increasing or decreasing at #x=0#?
1 Answer
Jan 7, 2018
Explanation:
#"to determine if f(x) is increasing/decreasing at x = a"#
#"differentiate and evaluate at x = a"#
#• " if "f'(x)>0" then f(x) is increasing at x = a"#
#• " if "f'(x)<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))^2larrcolor(blue)"quotient rule"#
#g(x)=-2x^3+x^2-2x-4rArrg'(x)=-6x^2+2x-2#
#h(x)=4x-2rArrh'(x)=4#
#f'(x)=((4x-2)(-6x^2+2x-2)-4(-2x^3+x^2-2x-4))/(4x-2)^2#
#rArrf'(0)=((-2)(-2)-4(-4))/4=5>0#
#"since " f'(x)>0" then f(x) is increasing at x = 0"#
graph{(-2x^3+x^2-2x-4)/(4x-2) [-10, 10, -5, 5]}
