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In: Math

Find the local maximum and minimum values and saddle point(s) of the function. If you have...

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(x, y) = x3 − 27xy + 27y3

Solutions

Expert Solution

critical point ( maximum ,minimum or saddle point ) of a function occur accordingly

     fx ( partial differentiation with respect to x ) = 0

and fy ( partial differentiation with respect to y ) = 0

then calculate D = fxx .fyy - (fxy)2 at the point where fx = 0 and fy=0

where fxx => partially differentiate fx with respect to x

          fyy => partially differentiate fy with respect to y

          fxy => partially differentiate fx with respect to y

If D>0 and fxx > 0 => relative minimum

If D>0 and fxx < 0 => relative maximum

If D<0 => saddle point

If D = 0 then no conclusion can be drawn


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