Question

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

Answer 1

$𝑓(𝑥,𝑦)=10𝑥^5−3𝑥^2𝑦^6−4𝑦^4$

Differentiate w.r.t x

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

Again differentiate w.r.t x-

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

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

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

Now differentiate w.r.t y-

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

Again differentiate w.r.t. y-

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

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

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