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**Deveno** along the line y = x we have to have dy/dx = -1 if the curve is symmetric, do we not?

think of it this way: suppose our curve WERE symmetric and $\displaystyle (x_1,y_1)$ is a point of the curve below y = x, and $\displaystyle (x_2,y_2)$ is the corresponding point of the curve above y = x.

if the curve is symmetric, we have $\displaystyle (x_2,y_2) = (y_1,x_1)$.

now, what is the slope between those two points? it's:

$\displaystyle \frac{y_2-y_1}{x_2-x_1} = \frac{x_1 - y_1}{y_1 - x_1} = -1$

now suppose a point where the curve crosses y = x is (a,a). then as $\displaystyle x_1 \to a$, the slope between the two points approaches dy/dx, but this slope is constantly -1.