Holiday Blog #3

Fort this blog I'm going to talk about how to use the First Derivative Test. To start off, you use it to find the relative/absolute extrema, max/mins, where the function is increasing/decreasing etc...In order to use it, you first take the derivative of the function you're given. Then you set the new equation equal to zero and solve for x to get your x-value(s). Then you set up intervals between negative infinity and infinity with those x-values. After that, you plug in numbers that would fall between those intervals (plugging them into the derivative) to determine if it is negative or positive at that specific x-value. If you get a negative number after plugging in, then that means the function is decreasing at that point. If you get a positive number after plugging in, then that means the function is increasing at that point. Thennnn, if you end up getting negative then positive after plugging in the two intervals, that means you have a min at that point. And if you get positive then negative that means you have a max at that point. Buttttttt, if you happen to get positive then positive, or negative then negative, that means there's nothing at that point.
Here's an example:

Ex 1) Determine the open intervals on which the function is increasing or decreasing
y = x^3/9 - 3x
*Okay first you need to take the derivative of this equation. To make it easier, I'm going to find a common denominator and then use the quotient rule to derive it
*So once you get a common denominator you should now have this equation: (x^3-27x)/9
*Now for the 1st step of the quotient rule you should have
(9)(3x^2-27)-[(x^3-27x)(0)]/(81)
*Simplifying that you get (27x^2-243)/81
*And to simplify that further, you can take out a 27 in the numerator leaving you with (x^2-9) in the numerator and 3 in the denominator
*So now you set the top of the fraction equal to zero (because the bottom doesn't matter)
x^2-9=0
x=3,-3
*Now you set up your intervals
(-infinity,-3)u(-3,3)u(3,infinity)
*Now plug in values for each interval...For the first interval you can plug in -4 (into the derivative) and you should get a positive number; then for the second interval you can plug in 0 and you should get a negative number; then for the last interval you can plug in 4 and you should get a postive number again
*So that means the function is increasing on (-infinity,-3)u(3,infinity)
And it's decreasing on (-3,3)
*And that also means you have a max at x=-3 and a min at x=3