# 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 ${\displaystyle f(x)}$ is increasing on an interval, if for two numbers ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$ in the interval ${\displaystyle x_{1} that ${\displaystyle f(x_{1}) is true.

A function ${\displaystyle f(x)}$ is decreasing on an interval, if for two numbers ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$ in the interval ${\displaystyle x_{1} that ${\displaystyle f(x_{1})>f(x_{2})}$ is true.

If a function ${\displaystyle f(x)}$ is continuous on a closed interval ${\displaystyle [a,b],}$ and differentiable on an open interval ${\displaystyle (a,b),}$ then the following applies:

1. If ${\displaystyle f'(x)>0}$ for all ${\displaystyle x}$ in ${\displaystyle (a,b),}$ then ${\displaystyle f(x)}$ is increasing on ${\displaystyle [a,b].}$

2. If ${\displaystyle f'(x)<0}$ for all ${\displaystyle x}$ in ${\displaystyle (a,b),}$ then ${\displaystyle f(x)}$ is decreasing on ${\displaystyle [a,b].}$

3. If ${\displaystyle f'(x)=0}$ for all ${\displaystyle x}$ in ${\displaystyle (a,b),}$ then ${\displaystyle f(x)}$ is constant on ${\displaystyle [a,b].}$

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 ${\displaystyle f(x)=x^{3}-{\frac {3}{2}}x^{2}.}$

First, check if the function is continuous for all ${\displaystyle x.}$

We can see the function exists for all ${\displaystyle x}$ therefore, it is continuous.

Second, find the critical numbers of ${\displaystyle f(x)}$ by using the derivative of the function.

Find the critical numbers by setting ${\displaystyle f'(x)=0.}$

${\displaystyle f'(x)=3x^{2}-3x}$

${\displaystyle 3x^{2}-3x=0}$

${\displaystyle x(3x-3)=0}$

${\displaystyle x=0,1.}$

Third, create intervals with your critical numbers.

Since we have two critical numbers, we will have three intervals. They are:

${\displaystyle -\infty

Fourth, determine if ${\displaystyle f'(x)}$ 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 ${\displaystyle -\infty ${\displaystyle 0 ${\displaystyle 1 Test Value ${\displaystyle x=-1}$ ${\displaystyle x={\frac {1}{2}}}$ ${\displaystyle x=2}$ Sign of ${\displaystyle f'(x)}$ ${\displaystyle f'(-1)=6}$ ${\displaystyle f'({\frac {1}{2}})={\frac {-3}{4}}}$ ${\displaystyle f'(2)=6}$ Increasing/Decreasing Increasing Decreasing Increasing

Lastly, determine if any relative maximums or minimums are present.

Since ${\displaystyle f'(x)}$ changes from increasing to decreasing to increasing, we can conclude that there is a relative maximum at ${\displaystyle x=0,}$ and a relative minimum at ${\displaystyle x=1.}$

#### Practice Problems

Find the relative extrema of the given functions.

${\displaystyle 1.f(x)=x^{2}-6x}$

${\displaystyle 2.f(x)=x^{4}-32x+4}$

${\displaystyle 3.f(x)=x+{\frac {1}{x}}}$