Find the intervals on which f is increasing or decreasing.

Question

(a)  Find the intervals on which f is increasing or decreasing.

(b)  Find the local maximum and minimum values of f.

(c)  Find the intervals of concavity and the inflection  points.

f (x) = x^4 - 2x^2 + 3

Answer 1

Step 1 :

(a)

Thu function .

Differentiate with respect to x:

Determination of critical points:

Since  is a polynomial it is continuous for all real numbers.

Thus, the critical points exist when .

Equate to zero:

The critical points are and the test intervals are .

Interval	Test Value	Sign of	Conclusion
			Decreasing
			Increasing
			Decreasing
			Increasing

Thus, The function is increasing on the intervals and .

And The function is decreasing on the interval and .

Solution : 

(a)

Increasing on the intervals and .

Decreasing on the interval and .

Answer 2

Step 1 :

(c)

Differentiate with respect to x:

Determination of concavity and inflection points :

Equate to zero:

Thus, the inflection points are split the intervals into and . .

Interval	Test Value	Sign of	Concavity
			Up
			Down
			Up

Thus, the graph is concave up in the interval and .

The graph is concave down in the interval .

Step 2 :

Inflection points :

Inflection points .

Solution :

Concave up on the interval and .

Concave down on the interval .

Inflection points .

Answer 3

Step 1 :  

(b)

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  .

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

Local minimum is .

Solution :

(b)

Local maximum is  .

Local minimum is .