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WZ
Numerade Educator

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Problem 79 Hard Difficulty

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

Answer

$$\begin{array}{l}
y=x \sin x \Rightarrow y^{\prime}=x \cos x+\sin x \Rightarrow y^{\prime \prime}=-x \sin x+2 \cos x, \quad y^{\prime \prime}=0 \Rightarrow 2 \cos x=x \sin x[\text { which is } y] \Rightarrow \\
(2 \cos x)^{2}=(x \sin x)^{2} \Rightarrow 4 \cos ^{2} x=x^{2} \sin ^{2} x \Rightarrow 4 \cos ^{2} x=x^{2}\left(1-\cos ^{2} x\right) \Rightarrow 4 \cos ^{2} x+x^{2} \cos ^{2} x=x^{2} \Rightarrow \\
\cos ^{2} x\left(4+x^{2}\right)=x^{2} \Rightarrow 4 \cos ^{2} x\left(x^{2}+4\right)=4 x^{2} \Rightarrow y^{2}\left(x^{2}+4\right)=4 x^{2} \text { since } y=2 \cos x \text { when } y^{\prime \prime}=0
\end{array}$$

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Video Transcript

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