Find the critical points of the following functions on the giv...

Question

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

ƒ(x) = sin 2x + 3 on [-π , π]

Accepted Answer

Below there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find the critical points of the function F of X is equal to cosine of 3 X minus 2 on the interval square brackets. Negative pi divided by 2, divided by 2 square bracket, identify the absolute maximum and minimum values. Awesome. So it appears for this particular problem we're asked to solve for three separate answers. We're asked to first solve for our critical points for our function F of X, so we're trying to find the critical points of F of X, and our other two answers we're trying to solve for is we're trying to identify the absolute maximum and minimum values. So those are our three main answers that we're ultimately trying to solve for. So with that said, First off, in order to find the critical points, we must first take the derivative of F of X. So we need to take the derivative of F of X is equal to the cosine of 3 x minus 2. So when we take the derivative, we will find that F of X, for this prime symbol denotes the first derivative of F of X, it's going to be equal to -3 multiplied by sin of 3 X. Fantastic. So now we need to set F of X to 0, and we will get that -3 multiplied by sin of 3 X is equal to 0, which therefore will imply as a result that sin of 3X is also equal to 0. Fantastic. So with that said, we now need to recall and note that the sine function equals 0 when the argument is an integer multiple of pi, meaning we can go ahead and write that 3 X is equal to n multiplied by pi. For where N is an element of the set integers. Awesome. So with that said, Using our little fancy Z, it looks like a letter Z, but it has like a little bit of a like a diagonal bar inside of it, so like a little fancy Z. So N with this thing that looks like resembles an E, so N in the symbol that resembles an E. And the symbol that resembles like a fancy Z, this state, so that's a fancy way of saying, like that's a way of writing that's the way to write how to say N is an element of the set integers. So the letter N. Is an element of the set integers. Awesome. So now with that said, we now need to rearrange this expression to isolate and solve for X by itself, and when we do, we will find that X is equal to N multiplied by pi divided by 3. And for integer values of N, let us find the possible critical points. So let us note that for N. is equal to 1, so when N is equal to -1, you should note that X is equal to negative pi divided by 3, and when N is equal to 0, X is equal to 0, and where N is equal to 1, X is equal to pi divided by 3. So with that said, moving right along, our next order of business, what we need to focus on or focus our attention on, is we need to solve for x. So we will find that when X is equal to 0, plus or minus pi divided by 3. So when solving for X, we need to find that X is equal to 0 and plus or minus pi divided by 3. So with that said, so when X is equal to negative pi divided by 2, We need to note that F of negative pi divided by 2 is going to be equal to cosine of 3 multiplied by negative pi divided by 2. These are plugging in a value of negative pi divided by 2 in for X. And then we need to subtract 2. So we take cosine of 3 multiplied by negative pi divided by 2 minus 2. And when we perform that calculation, we will find that it'll be equal to -2. And that X is equal to 0, we will find that F of 0 is going to be equal to cosine of 3 multiplied by 0 minus 2. Which is going to be equal to -1 when we perform that calculation. And and then now, our next step is we need to look at X is equal to pi divided by 3. So when X is equal to pi divided by 3. We can state that I'm just plugging it in. So when we plug in a value of pi divided by 3 in for X, we will get that cosine of 3 multiplied by pi divided by 3. minus 2 is going to be equal to -3. So, since you have a general idea of the pattern here, I'm not going to write out the the full equation anymore. I'm just going to plug in the value of X and then give you the final answer. So for when X is equal to negative pi divided by 3, that will mean that our final answer is going to be equal to -3, and once again we're plugging in a value of negative pi divided by 3 in for X. Fantastic. And now, for when X is equal to positive pi divided by 2, when we plug in that value for X, we will find that our final answer is going to be equal to -2. So now our next step is we need to evaluate F of X at these points and the endpoints of our intervals. So we need to pay attention to our interval, which, as we should recall and note, the square brackets indicate that it is a closed interval. So with that said, we need to take F of negative pi divided by 2 is equal to -2, and F of negative pi divided by 3 is equal to -3. And we need to state that F of 0 is equal to -1 and F of positive pi divided by 2 is equal to -2. And F of positive pi divided by 3 is equal to -3. So therefore, without a doubt, the absolute maximum will be -1 at X equals 0, and the absolute minimum will be -3 at X equals plus or minus pi divided by 3. All right, we did it, so to quickly recap. Let's note that our critical points, which you'll denote as C. So our critical points are going to be at X is equal to curly bracket, negative divided by 30 divided by 3, and then don't forget your curly bracket. And our F max, which once again is our, once again that's the value for the absolute maximum, it's going to be, and then we need to note that this is F max of 0, is going to be equal to -1. And let us also note that F min. Of negative pi divided by 3 is going to be equal to -3 and F min of positive pi divided by 3 will also be equal to -3. Fantastic. And those are our final answers. Hooray, we did it. So let's put a box around our final answers. So once again, to quickly recap, So just to reiterate, our final answers have to be without a doubt, so our critical points are X is equal to negative pi divided by 30, and pi divided by 3, and once again our absolute maximum at Negative, so so basically our absolute maximum at X equals 0 is gonna be -1, and our absolute minimums at -3, so far the answers, our absolute minimum is gonna be -3 at X equals plus or minus pi divided by 3. And that's it. Those are our final answers. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
ƒ(x) = sin 2x + 3 on [-π , π]

Key Concepts

Critical Points

Absolute Maximum and Minimum

Interval Notation

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