Find the local maximum and minimum values, saddle points of f(x,y) = x | Mathematics I 2078

Mathematics I20785 marksNumerical: local max min saddleMaximum and minimum values

Find the local maximum and minimum values, saddle points of $f(x,y) = x^4 + y^4 - 4xy + 1$ . [5]

Find the local maximum and minimum values, saddle points of $f(x,y) = x^4 + y^4 - 4xy + 1$. [5]

Function: $f(x,y) = x^4 + y^4 - 4xy + 1$ $fx = 4x^3 - 4y = 0 \Rightarrow y = x^3 \tag{1}$ $fy = 4y^3 - 4x = 0 \Rightarrow x = y^3 \tag{2}$ Substitute (1) into (2): $x = (x^3)^3 = x^9 \Rightarrow x^9 - x = 0 \Rightarrow x(x^8 - 1) = 0$ So $x = 0$ or $x = \pm 1$ . -

Function: $f(x,y) = x^4 + y^4 - 4xy + 1$ $$ fx = 4x^3 - 4y = 0 \Rightarrow y = x^3 \tag{1} $$ $$ fy = 4y^3 - 4x = 0 \Rightarrow x = y^3 \tag{2} $$ Substitute (1) into (2): $$x = (x^3)^3 = x^9 \Rightarrow x^9 - x = 0 \Rightarrow x(x^8 - 1) = 0$$ So $x = 0$ or $x = \pm 1$. -

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Find the local maximum and minimum values, saddle points of f(x,y) = x4 + y4 - 4xy + 1.