$(20$ pts $)$ Let

$(x,y)$ be in differniable function of two verialder

$x-3y4x=12$ be an equation of the tangent plane P to the graph of $x-f(x,y)$ at the point $A(1,2,2).$

Find the value of a and then find the maximum value of the directional derivative of $f(x,y)$ at $(L,a).$

$A(1,a,2)inp$;$x-3y4z=12=>1-3a8=12=>a=-1.$

The normal of $$ is vec$(n)=($:$1,-3,4$:$)||($:$f_x(1,-1),f_y(1,-1),-1$:$)$

$=>($:$1,-3,4$:$)=-4($:$f_x(1,-1),f_y(1,-1),-1$:$)$

$=>f_x(1,-1)=\frac{-1}{4}(20$ pts $)$ Let

$(x,y)$ be in differniable function of two verialder

$x-3y4x=12$ be an equation of the tangent plane P to the graph of $x-f(x,y)$ at the point $A(1,2,2).$

Find the value of a and then find the maximum value of the directional derivative of $f(x,y)$ at $(L,a).$

$A(1,a,2)inp$;$x-3y4z=12=>1-3a8=12=>a=-1.$

The normal of $$ is vec$(n)=($:$1,-3,4$:$)||($:$f_x(1,-1),f_y(1,-1),-1$:$)$

$=>($:$1,-3,4$:$)=-4($:$f_x(1,-1),f_y(1,-1),-1$:$)$

$=>f_x(1,-1)=\frac{-1}{4}$