Is #f(x)=(-x^2-5x-2)/(x^2+1)# increasing or decreasing at #x=-3#?
1 Answer
Increasing.
Explanation:
Find the value of the derivative at
To find the derivative of the function, use the quotient rule.
#f'(x)=((x^2+1)d/dx[-x^2-5x-2]-(-x^2-5x-2)d/dx[x^2+1])/(x^2+1)^2#
Each of these derivatives can be found through the power rule:
#d/dx[-x^2-5x-2]=-2x-5#
#d/dx[x^2+1]=2x#
Plugging these back in yields
#f'(x)=((x^2+1)(-2x-5)+(x^2+5x+2)(2x))/(x^2+1)^2#
Distribution and simplification of the numerator gives a derivative of
#f'(x)=(5x^2+2x-5)/(x^2+1)^2#
We can now find the value of the derivative at
#f'(-3)=(5(9)-6-5)/(9+1)^2=34/100=17/50#
Since this is
We can check the graph of
graph{(-x^2-5x-2)/(x^2+1) [-5, 5, -5.89, 3]}
