for f(x) = -x^3-6x^2-9x-2 determine all the critical points, test each interval and

Question

use the first and second derivative test to determine whee the graph is increasing or decreasing and which critical points are max and min values?

 

Answer 1

f(x)=-x³-6x²-9x-2.

f'(x)=-3x²-12x-9=0 at a turning point.

Divide through by -3: x²+4x+3=0=(x+3)(x+1). So there are turning points at x=-1, -3.

f(-1)=1-6+9-2=2; f(-3)=27-54+27-2=-2. Turning points at (-1,2) and (-3,-2).

f''(x)=-6x-12. f''(-1)<0 (maximum); f''(-3)>0 (minimum).

Max at (-1,2), min at (-3,-2). Therefore f(x) is decreasing (-∞,-3], increasing [-3,-1], decreasing [-1,∞).