Consider the following. f(x) = (3x + 2) / (x + 1) Find the...

Consider the following. f(x) = (3x + 2) / (x + 1) Find the critical numbers. (Hint: Check for discontinuities.) (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) x = Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) increasing decreasing

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00:01 So we have the function is a quotient 3x plus 2 over x plus 1 and so what i need to do is the quotient rule where the derivative of the top would be 3 you take leave the bottom alone minus a derivative of the bottom leave the top alone all over the denominator squared and what i would do is simplify the numerator just a bit because you end up with i'll just jot it up here 3x plus 3 minus 3x minus 2 and these 3x's will cancel so what you end up with top on top is 3 minus 2 is 1 and on the bottom you have x plus 1 squared and so what are your critical numbers your critical numbers are anywhere that the derivative is so let me write that critical numbers are where the derivative equals zero.

01:02 We'll look at the numerator can the derivative ever can the numerator ever equals zero? no, but it could also be where the derivative is undefined or does not exist if you heard that in class so that would be the denominator equal to zero and if you try to set the denominator equal to zero the only critical number would be x equals negative 1 but then if you try to see where the graph is increasing or decreasing so you're increasing if that prime is positive so greater than zero is increasing and so if i look at my number line, it's gonna put negative 1 on here real quick and we test numbers like x equals 0 or x equals negative 2 we're definitely gonna get a positive on top because that's what 1 is but the moment you plug in negative 2 plus 1 is negative 1 you square it gives you a positive as well so you're gonna get a positive on the left a negative 1 but if you plug in 0 you'll notice you'll also get a positive over a positive so where is the derivative positive and therefore increasing is on the interval from negative infinity to negative 1? union and with some of these problems, we'll just put a comment in between it but i'm gonna leave it as union you might put a comment though...

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