Find the local maximum and minimum values, saddle points of f(x,y) = x | Mathematics I 2077

Mathematics I20775 marksNumerical: Local extrema saddle pointsMaximum 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

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

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