Question #180127

If𝑓(𝑥,𝑦)=10𝑥5−3𝑥2𝑦6−4𝑦4.𝐹𝑖𝑛𝑑𝑓 (𝑥,𝑦),𝑓 (𝑥,𝑦), 𝑥𝑥 𝑥𝑦 𝑓 (1, −2), 𝑓 (1, 2)


1
Expert's answer
2021-04-29T17:13:50-0400

𝑓(𝑥,𝑦)=10𝑥53𝑥2𝑦64𝑦4𝑓(𝑥,𝑦)=10𝑥^5−3𝑥^2𝑦^6−4𝑦^4


Differentiate w.r.t x

f(x,y)x=50x4=6xy6f(x,y)_x=50x^4=6xy^6


Again differentiate w.r.t x-


f(x,y)xx=200x36y6f(x,y)_{xx}=200x^3-6y^6


f(1,2)xx=200(1)36(2)6=200384=184f(1,2)_{xx}=200(1)^3-6(2)^6=200-384=-184


f(1,2)xx=200(1)36(2)6=200384=184f(1,-2)_{xx}=200(1)^3-6(-2)^6=200-384=-184


Now differentiate w.r.t y-


f(x,y)y=18x2y516y3f(x,y)_y=-18x^2y^5-16y^3


Again differentiate w.r.t. y-


f(x,y)yy=90x2y448y2f(x,y)_{yy}=-90x^2y^4-48y^2


f(1,2)yy=90(1)2(2)448(2)2=1440192=1632f(1,2)_{yy}=-90(1)^2(2)^4-48(2)^2=-1440-192=-1632


f(1,2)yy=90(1)2(2)448(2)2=1140192=1632f(1,-2)_{yy}=-90(1)^2(-2)^4-48(-2)^2=-1140-192=-1632


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