Question

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) = 4(x2 + y2)ey2 − x2

local maximum value(s)=?

local minimum value(s) =?

saddle point(s) (x, y, f) =?

Answer 1

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

FUNCTION DOES NOT HAVE ANY MAXIMUM OR SADDLE POINTS