Find a unit vector in the direction of which the function f(x, y) = 6e^2 - x sin

Question

Find a unit vector in the direction of which the function f(x, y) = 6e^2 - x sin (y/3) (a) increases most rapidly, and (b) decreases most rapidly at the point (2, pi) A) (a) 1/2 Squareroot 7 (3 Squareroot 3 i - j), (b) 1/2 Squareroot 7 (-3 Squareroot 3 i + j) B) (a) 1/2 Squareroot 7 (3 Squareroot 3 i + j), (b) -1/2 Squareroot 7 (-3 Squareroot 3 i + j) C) (a) 1/2 Squareroot 7 (-3 Squareroot 3 i + j), (b) 1/2 Squareroot 7 (3 Squareroot 3 i - j) D) (a) -1/2 Squareroot 7 (3 Squareroot 3 i + j), (b) 1/2 Squareroot 7 (3 Squareroot 3 i + j) E) (a) 1/7 Squareroot 7 (3 Squareroot 3 i - j), (b) 1/7 Squareroot 7 (-3 Squareroot 3 i + j)

Explanation / Answer

given f(x,y) ===> 6 e^(2 - x)   sin (y/3)

f = {d/dx ( 6 e^(2 - x)   sin (y/3) ) , d/dy ( 6 e^(2 - x)   sin (y/3) ) }

    ===> { sin (y/3) 6 e^(2 - x ) d/dx ( 2 - x ) , 6 ( e^(2 - x) cos( y/3 ) d/dy ( y/3 ) }

=====> { - 6 e^(2 - x ) sin (y/3)   ,   2 e^(2 - x) cos( y/3 ) }
= {   - 6 e^(2 - 2 ) sin (/3)   ,   2 e^(2 - 2) cos( /3 )} ...... at (2 , )

==> { - 3 sqrt( 3 ) , 1)   ==> - 3 sqrt( 3 ) i + j
|| f || = (- 3 sqrt( 3 ))^2 + 1^2 ===>28 ==> 2 7

derivative in direction of fastest increase
u = {- 3 sqrt( 3 )/ 2 7 , 1/2 7}   ==>1/ 2 7   ( - 3 sqrt( 3 ) i + j )

Answer: max df/dt = 28 and u = {- 3 sqrt( 3 )/ 2 7 , 1/2 7}

derivative in direction of fastest decrease is

- u ==> - {- 3 sqrt( 3 )/ 2 7 , 1/2 7} ==> { 3 sqrt( 3 )/ 2 7 , - 1/2 7}   ==> 1/ 2 7   ( 3 sqrt( 3 ) i - j )

option C is correct

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