1. ## Derivative Graph Help

This is graph shows the derivative of a a continuous function f for [0,4]

A) for what values of x is f increasing? justify your answer

the x values from (0,2) and (3,4) are increasing because the derivative is positive, therefore the slope is positive so the x values are increasing.

B) For what values of x does f have its relative minimum value? Justify

there is a relative min at the x value of 3. this is because there is a 0 at 3 on the derivative graph. the derivative is negative to the left of 3 and positive to its right so the function goes down then comes back up making it a relative min.

C) For what value of x does f have its relative maximum value? Justify

there is no relative max because there are no zeros on the derivative graph that have positive values to its left and negative values to its right.

D) If f(1)=1, use your answers to A,B, & C to sketch the graph of f for [1,4]

2. Hello, rawkstar!

This is graph shows the derivative of a a continuous function $f$ for [0,4].

D) If $f(1)=1$, graph $f(x)$ on [0,4].

I believe the graph looks something like this.

Code:
        |               P
|               o               *
|          *    :*            * :
1 +       o       :   *       *   :
|     * :       :       *       :
|    *  :       :       :       :
|       :       :       :       :
|   *   :       :       :       :
- - + - - - + - - - + - - - + - - - + - -
|       1       2       3       4

At $x = 0$, the slope is about $m = 2\tfrac{1}{2}$

The graph passes through (1,1) and when it reaches $x = 2$ (point $P$)
. . the slope has decreased to $m = 2.$

Then the slope suddenly changes to $m = -1.$
The slope gradually increases to $m = 0$ at $x = 3$ (rel. minimum)
. . and continues increasing to $x = 4.$

We are told that the function is continuous.
So at point $P$, the two "halves" of the graph join up.

Hence, there is a relative maximum at $x = 2.$

3. thank you so much
but now i'm confused about question A
the first response says that the x values from (2,4) are increasing but the graph the second response shows that (2,3) is decreasing.

which is it?

4. Originally Posted by Soroban
Hello, rawkstar!

I believe the graph looks something like this.

Code:
        |               P
|               o               *
|          *    :*            * :
1 +       o       :   *       *   :
|     * :       :       *       :
|    *  :       :       :       :
|       :       :       :       :
|   *   :       :       :       :
- - + - - - + - - - + - - - + - - - + - -
|       1       2       3       4

There's something I don't quite understand about the graph.
In the graph of f' it seems that f'' is increasing on (0,3) and then at 3 it seems to hit a plateau and after that it decreases.

am I missing something?