$13$:$53$

MT$201$S$\_$HW$4\_$g$\_2024\_332132$a$...$ Done

MT$201$S Calculus $3$

$2024-25$ Semester $1$

Galatia Cleanthous

Homework $4$: Chapter $4$:

Extrema.

Answer all the questions; only one of them will be corrected.

Upload your answers as a single PDF file in Moodle.

Deadline: Friday December $6$th at $16$:$00.$

Good Luck!!

$(1)$ Consider the function $f(x,y)$ with formula

$f(x,y)=-x^2+xy-x-y^2+5y+1,(x,y)in{R}^2.$

$($a$)$ Show that its gradient vector equals

gradf$(x,y)=(-2x+y-1,x-2y+5).$

$($b$)$ Show that it has a unique critical point.

$($c$)$ Find the derivatives of order $2.$

$($d$)$ Find the Hessian determinant of $f.$

$($e$)$ Classify the above critical point.

$(2)$ Consider the function $f(x,y)$ with formula

$f(x,y)=6y^3+x^3+9y^{2}x-12x,(x,y)in{R}^2.$

$($a$)$ Find the partial derivatives $f_x,f_y,f_{},f_{xy}$ and $f_{yy}$

$($b$)$ Find and classify the critical points of $f.$

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