2. Given the function f(x) = x³ + x²

2. Given the function f(x) = x³ + x² - 8, answer the following: a. Is f(x) even, odd, or neither? Explain your reasoning. b. Over which open interval(s) is the function increasing? c. Over which open interval(s) is f(x) decreasing? d. What is the estimated x intercept of the function?

Transcript

00:01 All right, hello, in this question we're given this polynomial function f of x equals x cubed minus 3x squared minus 9x plus 3, and we're asked to find the max -mins inflection points and classify the concavity in increasing -decreasing regions.

00:11 So in order to do this we're going to start by taking our first derivative, we're going to have 3x squared minus 6x minus 9, and then my second derivative is going to be the derivative of that, which is going to be 6x minus 6.

00:22 I want to find my critical points, i want to set this equal to zero, so i can go ahead and factor 3 out of everything and just get rid of it because it's equal to zero, and i'm going to end up with this function here, and then i can factor that pretty easily.

00:36 I'm going to have x minus 3 and x plus 1, and that's going to equal zero there, and so x is going to be negative 1 or 3.

00:44 Those are going to be my two candidate points for maxes and mins.

00:47 So in order to test what they are, i'm going to do a first derivative test since i actually want to classify increasing -decreasing as well.

00:55 So i'm going to create regions, and i know that on these regions for x i'm going to have some sign for f prime of x, and it's probably going to change at negative 1 and 3.

01:06 So looking at f prime of x, this function here, if i plug in a very large negative number, the first term is going to dominate and it's squared, so it's going to be a positive number.

01:15 Then if i plug in something between negative 1 and 3, well, zero's in there, and so if i plug in zero i'll get negative 9, so i'm going to have a negative number there.

01:22 And then if i plug in a very large positive number, again this x squared term will dominate, so this is going to be positive.

01:27 So i have f of x is going to be increasing when the slope is positive, so this is my increasing, and this is increasing, and then this is decreasing.

01:38 And then i know that if i go from increasing to decreasing, that is going to correspond to the top, so that is going to be a max and that is going to be a min.

01:45 And i don't want to find just the x values, but i actually want to find the function value.

01:50 So i want to find what f of negative 1 and f of 3 are going to be.

01:54 If i plug in negative 1 into f, i'm going to get a value of 8, and that is going to be my max...

Question

2. Given the function f(x) = x³ + x² - 8, answer the following: a. Is f(x) even, odd, or neither? Explain your reasoning. b. Over which open interval(s) is the function increasing? c. Over which open interval(s) is f(x) decreasing? d. What is the estimated x intercept of the function?

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