Find the directions in which the function increases and decreases most rapidly at Po. Then find the derivatives of the function in these directionsf(x.y) x2 +xy+y2Po-3,-1)The direction in which the given function f(x,y) x xy+yincreases most rapidly at Po(-3, -1) is u(Type exact answers, using radicals as needed.)

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Asked Nov 2, 2019
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Find the direction f(x,y) increases and decreases most rapidly at point P and find the derivatives of the functions in these directions (f(x,y) and P included in the image)

Find the directions in which the function increases and decreases most rapidly at Po. Then find the derivatives of the function in these directions
f(x.y) x2 +xy+y2
Po-3,-1)
The direction in which the given function f(x,y) x xy+yincreases most rapidly at Po(-3, -1) is u
(Type exact answers, using radicals as needed.)
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Find the directions in which the function increases and decreases most rapidly at Po. Then find the derivatives of the function in these directions f(x.y) x2 +xy+y2 Po-3,-1) The direction in which the given function f(x,y) x xy+yincreases most rapidly at Po(-3, -1) is u (Type exact answers, using radicals as needed.)

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Expert Answer

Step 1

Consider the given function.

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f(x, y) x2 +xyy

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Step 2

To find the gradient of the function is the vector of partial derivatives.

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2x+y+0 af - 2x +y Ox At P(3,-1) =2(-3)+ (-1) axk--4) of -6-1 of exk-- =-7 Similarly g0+x+2y af x+2y At P,(-3,-1) =-3+2(-1) y\ --3-2 --1) --5 -3-1)

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Step 3

Therefore, the direc...

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af f Vf(x,y) axc f(x,y)-7i-5j

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