<s>[INST] Maximizing slope of a secant line Two points on the curve $$ y=\frac{x^3}{1+x^4}$$ have opposite $x$-values, $x$ and $-x$.  Find the points making the slope of the line joining them greatest.
Wouldn't the maximum slope of the secant line be with the max/min of the curve?
So $x=3^{1/4}$ and $x=-3^{1/4}$?
 [/INST] Replace $x^2=\tan(\alpha)$. We have $$\text{Slope} = \frac{\frac{x^3}{1+x^4}-\frac{-x^3}{1+x^4}}{2x}= \frac{x^2}{1+x^4} = \frac{\sin(2\alpha)}{2}\le\frac{1}{2}$$ with equality exactly when $\alpha=n\pi+\frac{\pi}{4}, x^2=1$
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