Find all points where the given function has any local extrema. Give the values

Question

Find all points where the given function has any local extrema. Give the values of any local extrema. Identify any saddly points. f(x,y) = 14x^2 - 2x^3 + 2y^2 + 4xy Find the local maxima. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. There are local maxima located at . There are no local maxima. Find the local value(s) of the local maxima. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The values of the local maxima are . There are no local maxima. Find the local minima. Select the correct choice below and necessary, fill in the answer box to complete your choice. There are local minima located at . There are no local minima. Find the value(s) of the local minima. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The values of the local minima are . There are no local minima. Find the saddle point(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. There are saddle points located at . There are no saddle points.

Explanation / Answer

fx = 28x - 6x^2+4y

fy = 4y+4x

fxy = 4

fxx =28-12x

fyy = 4

To find critical points fx = 0 and fy = 0 we get

fy = 0 ==> y = -x

Putting it in fx = 0 we get

28x -6x^2 -4x = 0

24x-6x^2 = 0

x = 0 , 4

critical points = (x,y) = (0,0) and (4,-4)

D (0,0) = fxx.fyy - (fxy)^2 =28*4-16 = 96>0 fx = 28> 0 minima

f(0,0) = 0

D(4,-4) = -20*4-16 =-96<0 saddle point

local minima at (0,0). Value = 0

no local maxima

saddle point (4,-4)

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