Question

 Find the local maximum and minimum values of  using both the First and Second Derivative Tests. Which  method do you prefer?

f (x) = 1 + 3x^2 - 2x^3.

Answer 1

Step 1:

The function

and

and

The critical points are and the test intervals are .

Step 2:

First derivative test:

Interval	Test Value	Sign of	Conclusion
			Decreasing
			Increasing
			Decreasing

is changes its sign from negative to positive, hence f  has a local minimum at .

Local minimum is  . 

is changes its sign from positive to negative, hence f  has a local maximum at . 

Local maximum is .

Solution:

Local minimum is  .

Local maximum is .

Answer 2

Step 1:

The function

Second derivative test:

Differentiate with respect to :

, curve is concave up, thus is a local minimum.

Local minimum is .

, curve is concave down, thus is a local minimum.

Local maximum is .

Solution:

Local minimum is  .

Local maximum is .