Mathematics

Problem

For the curve defined by

x^{2}+xy+y^{2}=7

, determine the

x

–coordinate(s) of all points where the tangent line is horizontal. a)

x=\pm\dfrac{2\sqrt{21}}{3}

x=\pm 3

x=\pm\sqrt{\dfrac{7}{3}}

x=\pm\sqrt{\dfrac{7}{2}}

Intuitive explanation

A horizontal tangent line means the slope is zero, which happens when $\frac{dy}{dx} = 0$ . Use implicit differentiation on the equation to find $\frac{dy}{dx}$ in terms of $x$ and $y$ , then set it equal to zero—this gives you a relationship between $x$ and $y$ . Substitute that relationship back into the original curve equation to solve for the actual $x$ -coordinates where this condition holds.

Intuitive explanation

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