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user65203

Because when $f'=g'=0$, the direction of the curve is no more defined. At such a point, the curve could change direction abruptly.

Example.

Because when $f'=g'=0$, the direction of the curve is no more defined. At such a point, the curve could change direction abruptly.

Because when $f'=g'=0$, the direction of the curve is no more defined. At such a point, the curve could change direction abruptly.

Example.

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user65203
user65203

If $f'(t)$ cancels, you have a maximum in $x$. (Or minimum or inflection.)

IfBecause when $g'(t)$ cancels$f'=g'=0$, you have a maximum in $y$. (Or minimum or inflectionthe direction of the curve is no more defined.)

And if both cancel, you can have At such a singular pointpoint, because $x$ and $y$ momentarily "stop varying" and can restart in anotherthe curve could change direction abruptly.

If $f'(t)$ cancels, you have a maximum in $x$. (Or minimum or inflection.)

If $g'(t)$ cancels, you have a maximum in $y$. (Or minimum or inflection.)

And if both cancel, you can have a singular point, because $x$ and $y$ momentarily "stop varying" and can restart in another direction.

Because when $f'=g'=0$, the direction of the curve is no more defined. At such a point, the curve could change direction abruptly.

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user65203
user65203

If $f'(t)$ cancels, you have a maximum in $x$. (Or minimum or inflection.)

If $g'(t)$ cancels, you have a maximum in $y$. (Or minimum or inflection.)

And if both cancel, you can have a singular point, because $x$ and $y$ momentarily "stop varying" and can restart in another direction.