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, local minimum, or saddle point. Confirm your results using a graphing utility. 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, local minimum, or saddle point. Confirm your results using a graphing utility.  f(x, y=(4x−1)2+(2y+4)2+1

zerogirlg16 · Accepted Answer

Step 1  Differentiate f(x,y)=(4x−1)2+(2y+4)2+1 partially with respect to x and y to obtain the values of fx and fy.  f(x,y)=(4x−1)2+(2y+4)2+1  fx=∂(4x−1)2+(2y+4)2+1∂x  =32x−8  fy=∂(4x−1)2+(2y+4)2+1∂y  =8(y+2)  Step 2  Equate fx and fy to 0 and solve for x and y.  fx=0  32x−8=0  x=832  =14  fy=8(y+2)  0=8(y+2)  y=−2  The critical point is (14, −2).  Step 3  Differentiate fx and fy partially with the respect to x and y respectively to obtain the value of fxx and fyy and differentiate fx with respect to y to obtain the value of fxy equate fxx and fyy to A and C respectively and fxy to B.  fxx=∂(32x−8)∂x  =32  A=32  fyy=∂(8(y+2))∂y  =8  C=8  fxy=∂(32x−8)∂y  =0  B=0  Step 4  Substitute the values of A,B,C in AC−B2 and obtain the value of the expression.  AC−B2=(32)(8)−02

Find the critical points of the following functions. Use the

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