suppose ( f^{prime prime} ) is continuous on ( (-infty, infty) ).\n(a) if ( f^{prime}(-3)=0 ) and ( f^{prime…

suppose ( f^{prime prime} ) is continuous on ( (-infty, infty) ).\n(a) if ( f^{prime}(-3)=0 ) and ( f^{prime prime}(-3)=-5 ), what can you say about ( f ) ?\n( \bigcirc ) at ( x=-3, f ) has a local maximum.\n( \bigcirc ) at ( x=-3, f ) has a local minimum.\n( \bigcirc ) at ( x=-3, f ) has neither a maximum nor a minimum.\n( \bigcirc ) more information is needed to determine if ( f ) has a maximum or minimum at ( x=-3 ).\n(b) if ( f^{prime}(-1)=0 ) and ( f^{prime prime}(-1)=0 ), what can you say about ( f ) ?\n( \bigcirc ) at ( x=-1, f ) has a local maximum.\n( \bigcirc ) at ( x=-1, f ) has a local minimum.\n( \bigcirc ) at ( x=-1, f ) has neither a maximum nor a minimum.\n( \bigcirc ) more information is needed to determine if ( f ) has a maximum or minimum at ( x=-1 ).

suppose ( f^{prime prime} ) is continuous on ( (-infty, infty) ).\n(a) if ( f^{prime}(-3)=0 ) and ( f^{prime prime}(-3)=-5 ), what can you say about ( f ) ?\n( \bigcirc ) at ( x=-3, f ) has a local maximum.\n( \bigcirc ) at ( x=-3, f ) has a local minimum.\n( \bigcirc ) at ( x=-3, f ) has neither a maximum nor a minimum.\n( \bigcirc ) more information is needed to determine if ( f ) has a maximum or minimum at ( x=-3 ).\n(b) if ( f^{prime}(-1)=0 ) and ( f^{prime prime}(-1)=0 ), what can you say about ( f ) ?\n( \bigcirc ) at ( x=-1, f ) has a local maximum.\n( \bigcirc ) at ( x=-1, f ) has a local minimum.\n( \bigcirc ) at ( x=-1, f ) has neither a maximum nor a minimum.\n( \bigcirc ) more information is needed to determine if ( f ) has a maximum or minimum at ( x=-1 ).

Answer

Brief Explanations:

Part (a)

  • Recall the second - derivative test: If (f^{\prime}(c)=0) and (f^{\prime\prime}(c)>0), then (f(x)) has a local minimum at (x = c). If (f^{\prime}(c)=0) and (f^{\prime\prime}(c)<0), then (f(x)) has a local maximum at (x = c).
  • Given (f^{\prime}(-3)=0) and (f^{\prime\prime}(-3)=-5<0). By the second - derivative test, at (x=-3), (f) has a local maximum.

Part (b)

  • The second - derivative test is inconclusive when (f^{\prime}(c) = 0) and (f^{\prime\prime}(c)=0).
  • For example, consider (y=x^{3}), (y^{\prime}=3x^{2}), (y^{\prime\prime}=6x). At (x = 0), (y^{\prime}(0)=0) and (y^{\prime\prime}(0)=0), and (y=x^{3}) has neither a local maximum nor a local minimum at (x = 0).
  • Consider (y=x^{4}), (y^{\prime}=4x^{3}), (y^{\prime\prime}=12x^{2}). At (x = 0), (y^{\prime}(0)=0) and (y^{\prime\prime}(0)=0), and (y=x^{4}) has a local minimum at (x = 0).
  • Consider (y=-x^{4}), (y^{\prime}=-4x^{3}), (y^{\prime\prime}=-12x^{2}). At (x = 0), (y^{\prime}(0)=0) and (y^{\prime\prime}(0)=0), and (y =-x^{4}) has a local maximum at (x = 0).

Answer:

(a) At (x=-3), (f) has a local maximum. (b) More information is needed to determine if (f) has a maximum or minimum at (x=-1).