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]}
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