<s>[INST] Need help finding smallest value of $x^2 + y^2$ I need to find the smallest value of $x^2 + y^2$ with the restriction $2x + 3y = 6$. This chapter focuses on the vertex formula. 
 [/INST] Using the Cauchy-Schwarz inequality, we have $6^2=(2x+3y)^2\leq (x^2+y^2)(2^2+3^2)$, which gives the minimum value of $x^2+y^2$ to be $\frac{36}{13}$.
Edit:Equality occurs for $\frac{x}{2}=\frac{y}{3}$.
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