# Increasing and Decreasing Functions

1. Jan 18, 2005

Hello all

$$y = \frac {1}{4} x^4 - \frac {2}{3}x^3 + \frac {1}{2}x^2 - 3$$

Find the exact intervals in which the function is

(a) increasing
(b) decreasing
(c) concave up
(d) concave down

Then find

(e) local extrema
(f) inflection points

So I found $$\frac {dy}{dx} = x^3 - 2x^2 + x$$.

I set $$\frac {dy}{dx} = x^3 - 2x^2 + x = 0$$ yielding $$x = 0 , 1$$ with 1 as a repeated root.

However after I find my critical points, how would I determine if the function is increasing or decreasing? Would I just choose a point less than, contained, and greater than the critical points and see if the sign changes from positive to negative, vice versa or not at all? If its - + - for example. the function is increasing than decreasing. But how do you know which intervals this happens in?

Okay, to determine concavity I find $$\frac {d^2y}{dx^2} = 3x^2 - 4x + 1$$. I set this equal to 0, find the value of x. Now how would I use these values to find if the function is concave up or down? I know that these points could be inflection points, but how do you know? I know if second derivative is negative, function is concave down and the same is true for the opposite. But what if the function changes concavity? How would you find the exact intervals where it is concave up or concave down?

Finally how do you find local extreme values? Do you just use the critical points? What is the difference in finding the relative extrema versus the local extrema?

Thanks a lot for any help!

PS: If you are given a graph, how would you determine relative and local extrema without being given the function?

Last edited: Jan 18, 2005
2. Jan 18, 2005

### MathStudent

Ok you kind of have the right ideas:

- You have the first derivative, how can this be interpreted graphically? How does this relate to increasing/ decreasing functions?

- What does the second derivative represent for a function of x? What does the second derivative tell you about the first derivative?

- What can you say about the above properties at a point that is a local max/min of a function?