let z=f(x,y) = exp( (-x^(3) / 3) + x - y^2 ) 1) find the critical points of f an

Question

let z=f(x,y) = exp( (-x^(3) / 3) + x - y^2 )

1) find the critical points of f and use the second derivative test or a function of two variables to determine the nature of the function (local maximum, local minimum or saddle point) at each critical point. If the function has a local max or min at a critical point, give the value of the function at that point.

2) determine the absolute maximum and minimum of f over the closed region bounded by the circle ( x - 1/2)^(2)+ (y^2) = 1

andthe points (x,y) where these extrema occur.

3) determine the absolute maximum and the minimum of f over the closed region bounded by the circle (x+1)^2 + y^2 = 1 a

and the points (x,y) where the extrema occur.

**please show work and steps on how to get to answers, to understand, & positive rating***

Explanation / Answer

f(x,y) = exp( (-x^(3) / 3) + x - y^2 )

Derivative= (x^2-1)(-( (-x^(3) / 3) + x - y^2 ))

local maximum = e^(2/3) at x=1,y=0

Local minimum= 0 at x=-34.0333, y= 120.216

2)Absolute max still = e^(2/3)

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