Mathematics

Mathematics

Problem

The equation of a curve is x2+y2=5x^{2}+y^{2}=5. Implicit differentiation shows that dydx\dfrac{dy}{dx} would be undefined at the point (3,0)(3,0). Explain, in one or two sentences, why (3,0)(3,0) is nevertheless not a critical point of the curve.
Intuitive explanation

The point (3,0)(3,0) doesn't actually lie on the curve x2+y2=5x^2 + y^2 = 5 (since 9+0=9≠59 + 0 = 9 \neq 5), so there's no tangent line to find there in the first place. A critical point must be a point on the curve where the derivative is undefined or zero—not just any point where the formula breaks down.

Intuitive explanation

The point (3,0)(3,0) doesn't actually lie on the curve x2+y2=5x^2 + y^2 = 5 (since 9+0=9≠59 + 0 = 9 \neq 5), so there's no tangent line to find there in the first place. A critical point must be a point on the curve where the derivative is undefined or zero—not just any point where the formula breaks down.

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