Determine the intervals on which f is increasing or decreasing and whether each of the critical points is a local maximum, minimum, or neither.f(x) = x^4 + 3x^3 + 3x^2 +1

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

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Given `f(x)=x^4+3x^3+3x^2+1` :

To determine where the function is increasing or decreasing we look at the first derivative. If the first derivative is positive on aninterval then the function is increasing on the interval; if negative then the function is decreasing.

A critical point (where the derivative is zero) will be a maximum if the function is increasing from the left and decreasing to the right; a minimum if decreasing from the left and increasing to the right.

`f'(x)=4x^3+9x^2+6x`

`4x^3+9x^2+6x=0`

`x(4x^2+9x+6)=0`

The only real zero is at x=0. Since the expression in the parantheses is always positive, we see that the derivative is negative for x<0, positive for x>0.

Thus the function decreases on `(-oo,0)` , increases on `(0,oo)` and the point (0,1) is a minimum.

The graph:

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