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

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

Explanation:

To determine if a function f(x) is increasing/decreasing at x = a we 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"#

To differentiate f(x) use the #color(blue)"quotient rule"#

#" If " 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)|)))#

#g(x)=-x^3+x^2-3x-4rArrg'(x)=-3x^2+2x-3#

#"and " h(x)=4x-2rArrh'(x)=4#

#rArrf'(x)#

#=((4x-2)(-3x^2+2x-3)-(-x^3+x^2-3x-4).4)/(4x-2)^2#

#rArrf'(0)=((-2)(-3)-(-4)(4))/(-2)^2#

#=22/4=11/2#

Since f'(0) > 0 , then f(x) is increasing at x = 0
graph{(-x^3+x^2-3x-4)/(4x-2) [-20, 20, -10, 10]}

answered by: Jim G.
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Is #f(x)=(-x^3+x^2-3x-4)/(4x-2)# increasing or decreasing at #x=0#?
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