Given f '(x) = (1 - x)(4 - x), determine the intervals on which f(x) is increasing or decreasing.

Question

solomonzelman · Answer

$\color{black}{\displaystyle f(x)}$ is decreasing iff $\color{black}{\displaystyle f'(x)<0}$.

solomonzelman · Answer

Explanation to the above:
   As you know $\color{black}{\displaystyle f'(x)}$ is the slope of $\color{black}{\displaystyle f(x)}$, and you also know that the function
   is decreasing if this slope (or $\color{black}{\displaystyle f'(x)}$) is negative. This is why we set  $\color{black}{\displaystyle f'(x)<0}$.

solomonzelman · Answer

With similar logic, if you want to start off from finding the intervals where $\color{black}{\displaystyle f(x)}$ is increasing, then you set $\color{black}{\displaystyle f'(x)>0}$.

solomonzelman · Answer

So, for example if I had:  $\color{black}{\displaystyle f'(x)=(x+3)(x-2)}$

Then, I set
$\color{black}{\displaystyle f'(x)>0}$
$\color{black}{\displaystyle (x+3)(x-2)>0}$

The above is true if both parenthesis are positive or both negative.
Therefore, $\color{black}{\displaystyle x\in(-\infty,\,-3)}$ and $\color{black}{\displaystyle x\in(2,\,\infty)}$ are the intervals where $\color{black}{\displaystyle f}$ is increasing.

You know that at $\color{black}{\displaystyle x=2,\,-3}$  the slope is 0.

So, therefore, you have the following interval where $\color{black}{\displaystyle f}$ decreases:
$\color{black}{\displaystyle x\in(-3,\,2)}$    (By excluding other values of the function.)

solomonzelman · Answer

So, the take-away here is:
$\color{blue}{\small \bf  \displaystyle [1] }$   $\color{black}{\displaystyle f(x) }$ is increasing on $\color{black}{\rm \displaystyle I }$   IFF   $\color{black}{\displaystyle f'(x)>0 }$ on $\color{black}{\rm \displaystyle I }$.
$\color{blue}{\small \bf  \displaystyle [2] }$   $\color{black}{\displaystyle f(x) }$ is decreasing on $\color{black}{\rm \displaystyle I }$   IFF   $\color{black}{\displaystyle f'(x)<0 }$ on $\color{black}{\rm \displaystyle I }$.