Suppose that fx5x2lnxx0 a list all the critical values of fx...

Suppose that f(x) = 5x^2 ln(x), x > 0 (A) List all the critical values of f(x). Note: If there are no critical values, enter 'NONE'. Critical values: (B) Use interval notation to indicate where f(x) is increasing. Note: Use 'INF' for ∞, '-INF' for −∞, and use 'U' for the union symbol. If there is no interval, enter 'NONE'. Increasing: (C) Use interval notation to indicate where f(x) is decreasing. Decreasing: (D) List the x values of all local maxima of f(x). If there are no local maxima, enter 'NONE'. x values of local maxima: (E) List the x values of all local minima of f(x). If there are no local minima, enter 'NONE'. x values of local minima: (F) Use interval notation to indicate where f(x) is concave up. Concave up: (G) Use interval notation to indicate where f(x) is concave down. Concave down: (H) List the x values of all the inflection points of f. If there are no inflection points, enter 'NONE'. x values of inflection points: (I) Use all of the preceding information to sketch a graph of f. Include all vertical and/or horizontal asymptotes. When you're finished, enter a "1" in the box below. Graph complete:

Transcript

00:01 Hello, so we have this function, f of x is 2x squared time l and x and basically from a to h is finding on critical point, interval increase, decrease, and local max, local mean, concave up, concave down, and inflection point is actually all about derivative.

00:16 So first, you need to find the first derivative, so f prime.

00:21 And in order to do that, what do you need to do? you need to do product rule because these are two function multiply together.

00:28 Okay.

00:29 So, uh, so, uh, uh, the function 2x squares what's the derivative 2x square and i'm going to be 4x multiply natural law x plus the second one is you keep the 2x square and derivative natural lock of x is 1 over x all right simplify this so 4x natural law x uh plus 2x right okay so the critical point is actually where this f prime equal 0 okay uh so you set this 4x natural of x plus 2x and you want to see equal to zero.

01:02 So you divide by two, so 2x, ln, x plus x equals zero.

01:07 Okay, so factor out on x.

01:08 So that will be 2 lnx plus 1 equals 0, right? so x will be 0 or the ln of x will be negative a half.

01:20 So you got x equal zero or x will be raising e to the pound negative one half.

01:27 You got two places for a critical point.

01:30 Okay.

01:31 Now for interval, increase and decreasing, actually, let's do it all at the same time, even local max and local mean.

01:37 Let's look at the table here.

01:39 So you've got a value zero here.

01:41 You got the value e to the negative one half here.

01:45 And then all you need to do is test the side of f prime.

01:49 For instance, you may want to test negative five.

01:52 And then you plug into right here, this is f -ram.

01:56 Remember this is f -ramm, right? so negative five.

01:59 Actually, because it's having, you don't have, sorry, my bad, you don't have this portion.

02:05 Why? because look at the domain, is x is bigger than zero, so you don't have that portion.

02:12 So all you need to do, let's test something here, zero to e to the negative one half.

02:18 You may need to put in your calculator to see what is e to the negative one half so that you can have kind of an estimate, right? this is approximately 0 .6 okay so how about try some number like one half which is 0 .5 you put it in here do you see a positive or negative to tam natural lock of 0 .5 plus one that is a negative so actually you get a negative here okay and then try something maybe try 1 here put in 1 so 2 l and 1 of 1 is 0 plus 1 so this is a positive.

02:59 So the way the function going like this because of a negative and then it's going like that because of a positive.

03:06 So you have the increase.

03:08 I'm sorry.

03:09 You have both increase and decrease.

03:11 So increase will be from e to the negative or half to infinity.

03:19 And then you can have an equal size on here because that's okay...

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