Question
Consider the function y 2 sin(*) 2 only o the interval (0,27) . Where decreasing? Enter exact values_and
Consider the function y 2 sin(*) 2 only o the interval (0,27) . Where decreasing? Enter exact values_ and
Answers
Find, if possible, the (global) maximum and minimum values of the given function on the indicated interval. $$f(x)=\sin ^{2} 2 x \text { on }[0,2]$$
Why Equal State of -2 signed data? From 0 to 2 pi. We are going to locate the absolute extreme. The derivative is one minus to co sign there. The function itself is never non differentiable. So the only possible critical numbers could be when that derivative is equal to zero. We have one minus two. Cosine. Theta equals zero. I'll subtract the one away which would give us negative too. Co signed data equals negative one. And if we divide by the -2, we get the statement. Co signed data is a half. There are a couple spots on the unit circle where that's true. The angle of pi over three And the angle of five pi over three. And those are inside the interval so we can use them to check for our absolute extreme. A Let's look at the y coordinates for the in point as well as the critical points. Okay, if we fell on zero for theta we would have zero minus to sign zero, that's zero minus zero or zero. If we fill in pi over three we would have pi over three minus to sign hi or three. And the sine of pi over three is radical 3/2. So filling that in will give me pi over three minus radical threat, Filling in five pi over three Would give us five pi over three minus to sign five pi over three and fought sign of five pi over three Is radical 3/2. Sorry, Negative Radical 3/2. So we end up with five pi over three plus Radical three. And finally if we fill in two pi we get to pi minus to sign two pi which is zero. So that's two past. So our smallest value is the one that is pi over three minus radical three. That actually is a negative number. So that's our minimum 5, 5 or three has the Y coordinate. That is the largest we convert those two decimal so that's our absolute max. Yeah,
Okay we'll find the maximums and minimums of this function on the interval 0-2 pi. So we've got to find the critical points so we've got to find the derivative. Okay remember this means for sign data Cubed -3. Cosine Data Squared. It is something cubed and something squared. So it's derivative is 12 something squared times the derivative of the something -6. Something to the one power times the derivative of the something. So 12 signs Square data. Co signed data plus six. Cosign Theta. Sine theta. I'm going to factor out a sign and co sign and a six And I get to sign data plus one. So sine of theta equals zero or co sin of theta equals zero Or sign of data equals -1/2. Yeah. All right sign of their equal zero on the unit circle. Where why is zero? Why is zero here and here. So data is zero Pi and to buy. Cosan if that is zero where X0 which is here and here. So if it is pi over two and three pi over two. Sign of data is negative in quadrant three and 4 sign is opposite over hypotenuse. Yeah So that's five or six is the reference angle. So 75 or six and 115 or six. Oh we got a lot of stuff to check here so we got zero. Bye two pi if I were to three point over to 75 or six and 11 5 or six. I don't want I'll put 11 5 or six at the top because I don't want to erase that Because I need it. Okay let's do 0 1st four times zero minus three times one. So that's my hopes that's minus oh well I'm sorry that's minus three a pie that zero minus three times negative one squared. That's also minus three 25 same as zero five or two. So that's four times one cubed minus zero. 3.5 or two. That's four times minus one cubed minus zero. Okay 75 or six. Got to write this time. Why? Of seven pi over six equals four times the sine of 75 or six is minus one half The coastline of 7. 5 or six is negative Negative Square Root of 3/2. So that's four times -18. eight -3 Times three. Force. That is -1 half -9. Force to -11. Force Which is -2 and 3/4. Okay and then 11 5 or six that's four times minus one half cubed minus three times positive square to 3/2 square. Which is going to give us the same answer because we had to square it in the first place. So -2 and three Force. Um Looks like this is the maximum and this is the minimum to me. Did I do those right pi over two is four times one cubed which is four and three by Red two is 4 yep. I think that's right
Oh, for this problem we see that we have a function that we can write ffx equal to x minus two sine of X. It gives us this graph right here. Now we can take the derivative of this function in order to find where exactly this graph has relative maximums and relative minimums. So this is going to be from 0 to 2 pi. So we'll limit it right here. We see that on this interval because we can't count zero and we can't count to pie. I'm instead we're going to look at the values in between. So we see that there is a relative minimum here. That doesn't count. We do have a relative maximum. We were out to a minimum right here at pi over three and we have a relative maximum right here at five pi over three. So going to those corresponding values, we see that our relative minimum is pi over three, negative point 685 and five Power 36.968
Yeah, we have problem number eight. Uh huh. This is why I called to science Correx. Why pollutions collects It means a foot calculating the function increasing and decreasing. We will be differentiating this. That is D Y by d x will be to sign X Cossacks to Synnex Cossacks. And we should, uh, just equated equal to zero for critical points. So cynics will busy row Cossacks will be zero so sign exit will be zero means x equal to end pie and cause actual V zero x equal toe to win plus one bye bye to So let us just get an idea about the values. Let us plug in an equal to zero in this case, actually be equal to zero and equal to one so pie to buy three pie And here, actually equal toe. If this is my way to by way too, creep, I way too Hi, Piper to it. Easy, etcetera. Okay, so these all are the critical points. Okay, so let us draw Mhm. This is why this is X. Yeah, no way. We have to draw the middle line, so let us draw the number Line X equal to zero. So we have X equal to zero Here, Zito. Bye bye to buy three pie by two Dubai It is it. So if he, uh, just take any value between zero to pi by two, it will be a little stick pie before so pie before signed by before is anything. So everything will be positive. And But if you plug in a diva Body X, that is at sign to works this is signed two x scientifics. So sign to work one so it will be equal to one which is positive. This must be negative, positive, negative and positive, So we can easily observe that. Okay, develop the ex changes that remember we're ascertaining the sign off. Divert the X that is after sex. So uh huh This is just like increasing this is Do you embody X here it is increasing. In this interval, it is decreasing, increasing, decreasing, increasing so increasing and decreasing. So at highway to we will be having maximum value at by minimum maximum minimum like this. Okay, Okay. So okay. We'll be writing affects equal to for minimum minimum maximum. We have to find increasing or decreasing function so increasing in between zero to pi by 20 to private too. Union? Yeah, by two three. Power to their power to buy 23 power to Uh huh in a D. C Like, um, the little power to fi toe three power to it will go on and it will go on. It will continue like this. So at this interval, it is increasing. Mhm. Remember, this is not the end of the interval. It will keep going. Keep going and decreasing will be with negative. So by way to toe by that 35222 pi In this interval, it is decreasing. Okay, so let us know. Get the graph. This is X. This is why one more thing. The grass cannot be negative because it is signed Square X so cannot be negative. So let us get critical points. What? We have got zero by were four If I were to Oh, no, not powerful. 052 by 352 by way to buy 3522 pi on a maximum value of science Corrects will be one only. Okay. Eso maximal leadoff Science Galaxies one. So let's draw a line of one. And at by way too. We have at highway to you had me maximum. So it will go like and at by it is a minimum. So they should go like this. It will keep going. Keep going. So this is the graph. Thank you.