Consider the function fxx2 5x 4x 2 a find the first...

Consider the function f(x)=(x^(2)-5x 4)/(x 2). a. Find the first derivative. f^(')(x)= ◻ b. List any critical values. ◻ syntax incomplete. c. Identify intervals of increase. d. Identify intervals of decrease. q, e. Find the second derivaitve. f^('')(x)= f. Based on parts b through e, f(x) has a maximum of y= q, when x= ◻ g. Based on parts b through e, f(x) has a minimum of y= ◻ when x= ◻ h. Use the second derivative to identify intervals where f(x) is concave up. ◻ Use the second derivative to identify intervals where f(x) is concave down. ◻ J. Use the second derivative to find any inflection points. ◻ k. State any vertical asymptotes I. State any slant asymptotes. ◻

Transcript

00:01 All right, so we're given f of x is equal to x squared minus 1 over x plus 2.

00:07 And it's a quotient, so we have to do the quotient rule.

00:10 So i'm just going to jump into that.

00:12 Take the derivative of the top, leave the bottom alone, minus the derivative of the bottom, leave the top alone, all over the denominator squared.

00:24 Now, i don't know if that's a good enough answer, but i'm going to go ahead and distribute.

00:27 So i'm looking at 2x squared plus 4x minus x squared plus 1.

00:37 So i might just rewrite my answer because 2x squared minus x squared is x squared plus 4x plus 1 all over x plus 2 squared.

00:50 So we don't have to worry about the denominator because it's undefined up here.

00:55 Here.

00:56 X cannot equal negative 2.

00:58 But you could have critical numbers in the numerator, which is not factorable.

01:02 So i would do the quadratic formula.

01:04 Negative b plus or minus the square root of b squared minus 4ac all over 2a, which i don't know if you just want to leave it or round your answers.

01:23 Let me go ahead and type that into my calculator because i get critical values of negative point two six eight but then if i did the minus one i get negative three point seven three two otherwise you might just leave it in simple stratical form negative four plus or minus it'll be the square root of 12 which you can break that down as 2 root 3 is the square root of 12 is equal to the square root of 4 times the square root of 3 but then you can actually divide each piece by two.

02:05 So that answer could be negative two plus or minus root three.

02:09 The same thing there.

02:12 I don't know.

02:14 So where are we increasing? i don't know if you have to give a big reason for this, but it's increasing from negative infinity to that first, this smaller one, negative 3 .732, and then again from the bigger one, negative 2 point, or sorry, 0 .268 to infinity, and then you need to be careful with your decreasing, because it does change signs at that negative 2, but negative 2 is undefined, so it doesn't technically increase there, or decrease 2, 6, 8.

02:53 So now they're asking you for the second derivative.

02:56 Again, if you don't like my answers, you can use the simple stratical form...

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