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.

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. $$ 1. f(x) = x^2 - 6x $$ $$ 2. f(x) = x^4 - 32x + 4 $$ $$ 3. f(x) = x + \frac {1}{x} $$