1. ## Curve Sketching

Did I do the question properly?

Sketch a graph of a function f(x) that is differentiable and that satisfies the following conditions:
• $f'(x) > 0$, when $x < -3$ and $x > 1$
• $f'(x) < 0,$ when $-3 < x < 1$
• $f'(-3) = 0$ and $f'(1) = 0$
• $f'(-3) = 5$ and $f(1) = 1$
Legend:
Purple is f(x)
Green is f'(x)

2. Hello,

No, it's false... the derivative is not positive in your sketch...
Actually, it's useless to draw f'(x). All that you have to remember is that if f'(x)>0, then f(x) will be increasing.

If f'(x)=0, there is a horizontal tangent line (here, it's g and g1 - see attachment), at A and B, which are turning points.

I've drawn a possible function, that satisfies the conditions

I hope it's clear enough... (sorry for the black line... couldn't do better)

3. Hello, Macleef!

I believe you have a typo in the problem . . .

Sketch a graph of a function $f(x)$ that is differentiable
and that satisfies the following conditions:

${\color{blue}[1]}\;\;f'(x) > 0\text{, when }x < \text{-}3\text{ and }x > 1$
${\color{blue}[2]}\;\;f'(x) < 0\text{, when } \text{-}3 < x < 1$
${\color{blue}[3]}\;\;f'(\text{-}3) = 0\text{ and }f'(1) = 0$
${\color{blue}[4]}\;\;{\color{red}f(\text{-}3) = 5} \text{ and } f(1) = 1$

[1] says the graph is rising when $x < -3\text{ and } x > 1$
[2] says the graph is falling when $-3 < x < 1$
Code:
          :           :
* :  *        :       *
*   :     *     :     *
*     :        *  :   *
*       :           : *
- - - - + - - - + - + - - - - -
-3       0   1

[3] says the slope is 0 at $x = -3\text{ and }x = 1.$
Those are extreme points (or turning points or stationary points).
Code:
        --o--         :
:           :
* :  *        :     *
*   :     *     :   *
*     :        *  : *
:         --o--
:           :
- - - - + - - - + - + - - - - -
-3       0   1

[4] tell us exactly where those extreme points are.
We can now sketch the curve.

Code:
       (-3,5)     |
o       |          *
*     *    |
*         *  |         *
*             |        *
*      *
*              |   o
|    (1,1)
- - - - + - - - + - + - - - - -
-3       |   1
|