# High School Calculus/The First Derivative Test
## The First Derivative Test

The definition of a derivatives tells us that a derivative is the slope of the tangent line at a point on the function.
Derivatives can also tell us if a function is decreasing or increasing at a point.
A function $f(x)$ is increasing on an interval, if for two numbers $x_{1}$ and $x_{2}$ in the interval $x_{1}<x_{2},$ that $f(x_{1})<f(x_{2})$ is true.
A function $f(x)$ is decreasing on an interval, if for two numbers $x_{1}$ and $x_{2}$ in the interval $x_{1}<x_{2},$ that $f(x_{1})>f(x_{2})$ is true.
If a function $f(x)$ is continuous on a closed interval $[a,b],$ and differentiable on an open interval $(a,b),$ then the following applies:
1. If $f'(x)>0$ for all $x$ in $(a,b),$ then $f(x)$ is increasing on $[a,b].$
2. If $f'(x)<0$ for all $x$ in $(a,b),$ then $f(x)$ is decreasing on $[a,b].$
3. If $f'(x)=0$ for all $x$ in $(a,b),$ then $f(x)$ is constant on $[a,b].$
In the last section, we learned about absolute minimums/maximums. Inside a function, other extrema, known as relative extrema, can exist.
The relative extrema of a function are points on a function that are lower or higher than all of the points near them. Such points create "hills" or "valleys" within a given function.
Relative extrema occur at points on a function where the derivative at that point changes from increasing to decreasing, or decreasing to increasing.
If the derivative changes from increasing to decreasing, that point is known as a relative maximum.
If the derivative changes from decreasing to increasing, that point is known as a relative minimum.
By finding the relative extrema of a function, you can then calculate whether or not those extrema are relative minima or maxima using the derivative of the function at those points.
Relative extrema are always critical points of a function.

### Example

Find the relative extrema of $f(x)=x^{3}-{\frac {3}{2}}x^{2}.$
First, check if the function is continuous for all $x.$
We can see the function exists for all $x$ therefore, it is continuous.
Second, find the critical numbers of $f(x)$ by using the derivative of the function.
Find the critical numbers by setting $f'(x)=0.$
$f'(x)=3x^{2}-3x$
$3x^{2}-3x=0$
$x(3x-3)=0$
$x=0,1.$
Third, create intervals with your critical numbers.
Since we have two critical numbers, we will have three intervals. They are:
$-\infty <x<0,0<x<1,1<x<\infty .$
Fourth, determine if $f'(x)$ is increasing or decreasing over each interval. Do this by evaluating a test number within each interval.
In most cases, it is beneficial to create a table to arrange the present data.
| Interval | $-\infty <x<0$ | $0<x<1$ | $1<x<\infty$ |
| Test Value | $x=-1$ | $x={\frac {1}{2}}$ | $x=2$ |
| Sign of $f'(x)$ | $f'(-1)=6$ | $f'({\frac {1}{2}})={\frac {-3}{4}}$ | $f'(2)=6$ |
| Increasing/Decreasing | Increasing | Decreasing | Increasing |
Lastly, determine if any relative maximums or minimums are present.
Since $f'(x)$ changes from increasing to decreasing to increasing, we can conclude that there is a relative maximum at $x=0,$ and a relative minimum at $x=1.$

#### Practice Problems

Find the relative extrema of the given functions.
$1.f(x)=x^{2}-6x$
$2.f(x)=x^{4}-32x+4$
$3.f(x)=x+{\frac {1}{x}}$