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

Question

Samantha Adkison · Accepted Answer

decreasing at x = -2

We evaluate f'(a) to find out if a function is increasing or decreasing at x = a. • At x = a, f(x) is increasing if f'(a) > 0. • At x = a, f(x) is decreasing if f'(a) < 0. differentiate f(x) using the #color(blue)" quotient rule " # If # f(x) = (g(x))/(h(x)) " then " f'(x) =(h(x).g'(x) - g(x).h'(x))/(h(x))^2 # #"--------------------------------------------------------------"# g(x) #-2x^3+4x^2-x-2 rArr g'(x) = -6x^2+8x-1 # h(x) = x + 3 → h'(x) = 1 #"----------------------------------------------------------"# Substitute these values into f'(x) f'(x) #=((x+3)(-6x^2+8x-1)-(-2x^3+4x^2-x-2).1)/(x+3)^2# and f'(-2)#=(1.(-24-16-1)-(16+16+2-2))/1# = (-41-32) = -73 f(x) is decreasing at x = -2 because f'(-2) < 0. The graph is represented as (-2x^3+4x^2-x-2)/(x+3) [-10, 10, -5, 5]}.

Isabella Ackerman · Answer

undefined To determine if the function f(x) = (-2x^3 + 4x^2 - x - 2) / (x + 3) is increasing or decreasing at x = -2, we need to analyze the sign of the derivative of the function at x = -2. The function is increasing if the derivative is positive at x = -2, and it is decreasing if the derivative is negative at x = -2.

After finding the derivative and evaluating it at x = -2, if the derivative is positive, the function is increasing at x = -2. If the derivative is negative, the function is decreasing at x = -2.

Therefore, we differentiate the function f(x) = (-2x^3 + 4x^2 - x - 2) / (x + 3) and evaluate it at x = -2 to determine if it's increasing or decreasing at that point.