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# Determine and classify all critical points of the function: f$x, y$ = y^4 − x^3 − 2y^2 + 3x.

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## Answer to a math question Determine and classify all critical points of the function: f$x, y$ = y^4 − x^3 − 2y^2 + 3x.

Maude
4.7
1. Calcular as derivadas parciais da função:
\frac{\partial f}{\partial x} = -3x^2 + 3
\frac{\partial f}{\partial y} = 4y^3 - 4y

2. Resolver o sistema para encontrar os pontos críticos:
-3x^2 + 3 = 0 \Rightarrow x = 1 \; \text{ou} \; x = -1
4y^3 - 4y = 0 \Rightarrow y = 0 \; \text{ou} \; y = 1 \; \text{ou} \; y = -1

3. Encontrar todos os pontos críticos:
$1, 0$, $1, 1$, $1, −1$, $−1, 0$, $−1, 1$, $−1, −1$

4. Calcular a matriz Hessiana:
H$f$ = \begin{bmatrix}-6x & 0 \\0 & 12y^2 - 4 \end{bmatrix}

5. Avaliar a Hessiana nos pontos críticos para classificá-los:
$1, 0$: \; D = 24 > 0 \; $\text{mas} \; H_{11} < 0 \rightarrow \text{sela}$
$1, 1$: \; D = 48 > 0 \; $\text{e} \; \lambda_1 < 0, \lambda_2 < 0 \rightarrow \text{máximo relativo}$
$1, -1$: \; D = 48 > 0 \; $\text{e} \; \lambda_1 < 0, \lambda_2 < 0 \rightarrow \text{máximo relativo}$
$-1, 0$: \; D = 24 > 0 \; $\text{mas} \; H_{11} < 0 \rightarrow \text{sela}$
$-1, 1$: \; D = 48 > 0 \; $\text{e} \; \lambda_1 > 0, \lambda_2 > 0 \rightarrow \text{mínimo relativo}$
$-1, -1$: \; D = 48 > 0 \; $\text{e} \; \lambda_1 > 0, \lambda_2 > 0 \rightarrow \text{mínimo relativo}$

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