Is f(x)=(-2x^3+4x^2-x-2)/(x+3) increasing or decreasing at x=-2?
1 Answer
decreasing at x = -2
Explanation:
To determine if a function 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
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
Since f'(-2) < 0 then f(x) is decreasing at x = -2
graph{(-2x^3+4x^2-x-2)/(x+3) [-10, 10, -5, 5]}