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The graphs of $y=3-x^2+x^3$ and $y=1+x^2+x^3$ intersect in multiple points. Find the maximum difference between the $y$-coordinates of these intersection points.

Mar 28, 2018

#1
+3 This graph should help:

Mar 28, 2018
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The graphs of $y=3-x^2+x^3$ and $y=1+x^2+x^3$ intersect in multiple points. Find the maximum difference between the $y$-coordinates of these intersection points. $$\begin{array}{|rcll|} \hline 3-x^2+\not{x^3} &=& 1+x^2+ \not{x^3} \\ 3-x^2 &=& 1+x^2 \quad & | \quad -1-x^2 \\ 3-x^2 -1-x^2 &=& 0 \\ 2-2x^2 &=& 0 \\ 2(1-x^2) &=& 0 \quad & | \quad : 2 \\ 1-x^2 &=& 0 \\ x^2 &=& 1 \quad & | \quad \sqrt() \\ x &=& \pm\sqrt{1} \\\\ x_1 &=& 1 \\ y_1 &=& y(1) \\ &=& 1+1+1 \\ \mathbf{y_1 } &\mathbf{=}& \mathbf{3} \\\\ x_2 &=& -1 \\ y_2 &=& y(-1) \\ &=& 1-1+1 \\ \mathbf{y_2} &\mathbf{=}& \mathbf{1} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline y_1 - y_2 &=& 3-1 \\ &=& 2 \\ \hline \end{array}$$

The maximum difference between the  y-coordinates of these intersection points is 2  Mar 28, 2018