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Problem 37 Hard Difficulty

Use Newton's method to find the coordinates of the inflection point of the curve $ y = x^2 \sin x $, $ 0 \leqslant x \leqslant \pi $, correct to six decimal places.

Answer

inflection point at approximately $(1.519855,2.306964)$

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Video Transcript

question 37. We are looking at a function of why in court ex clear sign. So I took my group. Why? Ride and I had to use the, um Brad, It rule crying G plus g prime f So I have X sine X plus x where co sign IX and then I'm gonna take my Why Devil Crying and again I have to apply the chain of the product rule for both of these. No, I have to sign X plus two x coz I necks And then I'm looking at the second part plus two x coz my next And then when I take the g prime get a negative sign So it x weird sign Thanks. Combining my light terms I am going to get combine these two I have to sign of X plus we're ex sign Thanks My ex weird sign of necks And now I want my people why people cried So I take the derivative of to sign next which is just to sign Thanks. Then I have to do my product rule or four ex co scientific. So I get four co sign X minus were X sign of minus No on the last piece that minus two x sign of X minus X weird. Who? Sign X Combining my like terms. I am going to get six co sign of X minus six x sign of X minus X square Code signing Vicks. So I'm using, um, 1.5 is my approximate starting point. So I am putting in, um X minus a devil prime over F triple prime. And in your graphing calculator, if you put these in like l won our story, Why, 1123 white, four. And then you can just go to bars. And in white five, you can just have the wife. Why Double Brian? 1/4 bye bye. Why Triple crime? It's a lot easier on that. Right Did. So I got, um when I plugged in 1.5 into this right here, one x one, of course, people. Approximately 1.5. My first inflection point. And then I got accepted to 1.520092 ex of three. One point 519 855 And I also got that for, except for. So that becomes my ex value. And I put my ex value into my original wife function. And when I do that, when I do, uh, 1.519855 I get my value. 2.3 06 90 Yes, it's 90 six, Yeah.