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Question

# Find all points (x, y) at which the tangent plane to the graph of $z=x^3$ is horizontal.

Solution

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$z=f\left( x,y\right)=x^{3}$

\begin{align*} \nabla f\left(x,y \right)&=\dfrac{\partial f\left(x,y \right)}{\partial x}\hat{i} +\dfrac{\partial f\left(x,y \right)}{\partial x}\hat{j}\\ &=3x^{2}\hat{i} +0\hat{j} \\ &=3x^{2}\end{align*}

Tangent plane is horizontal then $z$ constant

\begin{align*} \nabla f\left( x,y\right)&=\left\langle 0,0\right\rangle\\ \left\langle 3x^{2},0 \right\rangle&=\left\langle 0,0\right\rangle\\ x&=0\end{align*}

we conclude ,the tangent plane is horizontal at point $\left(0,0\right)$

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