Consider the equation below. (If an answer does not exist,...

Consider the equation below. (If an answer does not exist, enter DNE.) f(x) = 3 sin(x) + 3 cos(x), 0 ? x ? 2? (a) Find the interval on which f is increasing. (Enter your answer using interval notation.) Find the interval on which f is decreasing. (Enter your answer using interval notation.) (b) Find the local minimum and maximum values of f. local minimum value local maximum value (c) Find the inflection points. (x, y) = ( ) (smaller x-value) (x, y) = ( ) (larger x-value) Find the interval on which f is concave up. (Enter your answer using interval notation.) Find the interval on which f is concave down. (Enter your answer using interval notation.)

Transcript

00:01 Okay, we need to find the critical numbers.

00:03 So let's take the derivative.

00:05 3 cosine x minus 3 sine x equals to zero.

00:12 So technically what we have here, cosine of x is equal to sine of x.

00:18 That means which values sine and cosine are same.

00:24 Meaning they are both positive here and they are both negative there.

00:27 So we know it's 45 pi over 4 and then this one is 5 pi over 4.

00:34 225.

00:35 Okay, and so those are our critical points.

00:44 I'm going to go ahead and find also the second derivative real quick.

00:47 Okay, for concavity.

00:50 So second derivative is the derivative of first derivative.

00:53 Negative 3 sine x times negative 3 cosine x is equal to zero.

00:58 So in this case, negative sine x is equal to cosine x.

01:04 So now the opposite sides, meaning when sine and cosine are oppositely equal.

01:15 So again, pi over 4, but this is 3 pi over 4 or 135 and then this is 7 pi over 4 or 315 degrees.

01:29 So we can make a big table like this and i'm going to put a couple of things here so you can easily fill it up.

01:41 So because this is also a closed interval problem, i'm going to include the end points.

01:48 Zero to 2 pi.

01:50 So this is y prime.

01:53 This is y double prime.

01:56 And this is the function value.

01:57 Okay, so and critical numbers we have is right here we have pi over 4 and we have here 5 pi over 4.

02:11 Okay, so let's make it room for like that and actually room for like that right here...

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