Hello, Macleef!
I believe you have a typo in the problem . . .
Sketch a graph of a function $\displaystyle f(x)$ that is differentiable
and that satisfies the following conditions:
$\displaystyle {\color{blue}[1]}\;\;f'(x) > 0\text{, when }x < \text{}3\text{ and }x > 1$
$\displaystyle {\color{blue}[2]}\;\;f'(x) < 0\text{, when } \text{}3 < x < 1$
$\displaystyle {\color{blue}[3]}\;\;f'(\text{}3) = 0\text{ and }f'(1) = 0$
$\displaystyle {\color{blue}[4]}\;\;{\color{red}f(\text{}3) = 5} \text{ and } f(1) = 1$
[1] says the graph is rising when $\displaystyle x < 3\text{ and } x > 1$
[2] says the graph is falling when $\displaystyle 3 < x < 1$
Code:
: :
* : * : *
* : * : *
* : * : *
* : : *
    +    +  +     
3 0 1
[3] says the slope is 0 at $\displaystyle x = 3\text{ and }x = 1.$
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
