Is f(x)=(-2x^3+x^2-2x-4)/(4x-2) increasing or decreasing at x=0?

1 Answer
Jan 7, 2018

"increasing at x = 0"

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]}