Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum,or a saddle point. If the Second Derivative Test is inconclusive,determine the behavior of the function at the critical points. f(x, y)=(4x−1)2+(2y+4)2+1

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Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum,or a saddle point. If the Second Derivative Test is inconclusive,determine the behavior of the function at the critical points.  f(x, y)=(4x−1)2+(2y+4)2+1

Fatema Sutton · Accepted Answer

Step 1 Given function is f(x)=(4x−1)2+(2y+4)2+1=16x2+4y2−8x+16y+18 We find critical points: fx=∂f∂x=32x−8 fy=∂fdy=8y+16 Setting these equal to zero, 32x−8=0, ⇒x=14 8y+16=0, ⇒y=−2 So, the critical point is {(14, −2}) Now we find the second derivatives, f×=32 fyy=8 fxy=0 So, D=f××fyy−(fxy)2 For {(14, −2}) D=32×8−0=256&amp;gt;0 Since D&amp;gt;0 and f×{(14, −2})=32&amp;gt;0, so the obtained critical point is a local minimum.

Find the critical points of the following functions.Use the Second

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