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Show that the inflection points of the curve $y=x \sin x$ lie on the curve $y^{2}\left(x^{2}+4\right)=4 x^{2}.$

SEE EXPLANATION

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

Derivatives

Differentiation

Applications of the Derivative

Campbell University

University of Michigan - Ann Arbor

Idaho State University

Lectures

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for this program This right out, right up a prime for us. So whatever crying cause to twice off consign x minus X science. So off the inflection point Satisfied wipe double prime because zero reach muse two times Kasai X equals two x science. So I label this by equation one. Um So the conclusion is that if so, I should say that, um all create your point. So are you Inflection point satisfies equation one. Now we want verified off This points our own a curve. So for the given curve the right hand side you coast to why square plus at times x squared plus fork And the record that why equals two x times Sile fix the White Square equals two x squared times Thanks to square times X we're proud for Okay, No, also compound with the question one. We know that for any inflection point X times Sigh X equals two twice off course sex. So it's we repairs the 1st 4th term here by four times co sign X square and we keep the second time on change. Okay, now again, from your question one, we have an expression for X, so x will be, uh, twice off course I X over. Son X. Um, I just divided science for both sides and the we replaced this eggs into on this right inside the question So we have four times cause i x squared times x square So it's four times cause I x square divided by Sykes Square plows full. Okay, so now we will use the trig identity Here, Find out a commodity, Nominator We have sine x square Onda for the numerator will be four time school sine x square class four times sign X square, right, which is off. So for this numerator by the trick, identity is just for so we have four times close i x square times one over sigh X square, which is so there's another four here which is four times Let me see so four times what times cause I x square over Sykes Square, which is exactly for X because X equals 2 to 2 times cause an X over a sex. So we verified This equals this is in fact, this is the left inside. So right hand side across 11 said that so we finish the proof you

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