# CALCULUS QUESTION PLEASE HELP Find f(x)= -x^3 - 6x^2 - 9x -2 determine all the critical pointsALSO test each interval and use the first and second derivative test to determine where the graph is...

Find f(x)= -x^3 - 6x^2 - 9x -2 determine all the critical points

ALSO test each interval and use the first and second derivative test to determine where the graph is increasing or decreasing and which critical points are max and min values

Asked on by mcoach222

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

Given `f(x)=-x^3-6x^2-9x-2` , determine critical points.

Critical points are located at the points where `f'(x)=0` or does not exist. Here, f(x) is infinitely differentiable, so we need only find the zeros.

(1) `f'(x)=-3x^2-12x-9=-3(x^2+4x+3)=-3(x+1)(x+3)`

Setting `f'(x)=0` yields `-3(x+1)(x+3)=0=>x=-1,-3` .

So the critical points are x=-1 and x=-3.

(2) We test points in the intervals `(-oo,-3),(-3,-1)(-1,oo)`

`f'(-4)=-9<0` Thus the function is decreasing on `(-oo,-3)`

`f'(-2)=3>0` Thus the function is increasing on `(-3,-1)`

`f'(0)=-9<0` Thus the function is decreasing on `(-1,oo)`

(3) X=-3 is a relative minimum (since the function decreases from the left and increases to the right), x=-1 is a relative maximum (since the function increases from the left and decreases to the right)

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