the function ( f ) has continuous second derivatives, and a critical point at ( (1,8) ). suppose (…

the function ( f ) has continuous second derivatives, and a critical point at ( (1,8) ). suppose ( f_{xx}(1,8)=-2, f_{xy}(1,8)=-2, f_{yy}(1,8)=2 ). then at the point ( (1,8) ):\na. there is a local minimum\nb. there is a saddle point\nc. the second derivative test is inconclusive\nd. there is a local maximum\ne. none of the above

the function ( f ) has continuous second derivatives, and a critical point at ( (1,8) ). suppose ( f_{xx}(1,8)=-2, f_{xy}(1,8)=-2, f_{yy}(1,8)=2 ). then at the point ( (1,8) ):\na. there is a local minimum\nb. there is a saddle point\nc. the second derivative test is inconclusive\nd. there is a local maximum\ne. none of the above

Answer

Explanation:

Step1: Calculate the discriminant (D)

The formula for the discriminant (D) of a function (f(x,y)) at a critical point ((a,b)) is (D = f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^{2}). Given (f_{xx}(1,8)=- 2), (f_{xy}(1,8)=-2), (f_{yy}(1,8)=2). Substitute these values into the formula: (D=(-2)\times2-(-2)^{2}) (D=-4 - 4) (D=-8)

Step2: Use the second - derivative test

The second - derivative test states that:

  • If (D>0) and (f_{xx}(a,b)>0), then (f(a,b)) is a local minimum.
  • If (D>0) and (f_{xx}(a,b)<0), then (f(a,b)) is a local maximum.
  • If (D<0), then ((a,b)) is a saddle point.
  • If (D = 0), the test is inconclusive.

Since (D=-8<0)

Answer:

B. There is a saddle point