the function ( f(x) ) is continuous on ( (-infty,infty) ). use the given information to sketch the graph of…

the function ( f(x) ) is continuous on ( (-infty,infty) ). use the given information to sketch the graph of ( f ).\n( f(-5)=25,f(0)=0,f(5)=-25 )\n( f(-5)=0,f(0)=0,f(5)=0 );\n( f(x)>0 ) on ( (-infty,-5) ) and ( (5,infty) );\n( f(x)<0 ) on ( (-5,0) ) and ( (0,5) )\nchoose the correct graph of ( f ) below.

the function ( f(x) ) is continuous on ( (-infty,infty) ). use the given information to sketch the graph of ( f ).\n( f(-5)=25,f(0)=0,f(5)=-25 )\n( f(-5)=0,f(0)=0,f(5)=0 );\n( f(x)>0 ) on ( (-infty,-5) ) and ( (5,infty) );\n( f(x)<0 ) on ( (-5,0) ) and ( (0,5) )\nchoose the correct graph of ( f ) below.

Answer

Explanation:

Step1: Analyze the sign of the first - derivative

  • When (f^{\prime}(x)>0) on ((-\infty,-5)) and ((5,\infty)), the function (f(x)) is increasing on these intervals.
  • When (f^{\prime}(x)<0) on ((-5,0)) and ((0,5)), the function (f(x)) is decreasing on these intervals.

Step2: Analyze the critical points

  • Since (f^{\prime}(-5) = 0), (x=-5) is a critical point. And because the function changes from increasing ((x < - 5)) to decreasing ((-5<x<0)), (f(-5)=25) is a local maximum.
  • Since (f^{\prime}(5)=0), (x = 5) is a critical point. And because the function changes from decreasing ((0<x<5)) to increasing ((x>5)), (f(5)=-25) is a local minimum.
  • (f^{\prime}(0) = 0), but the function is decreasing on both sides of (x = 0) (from (-5<x<0) to (0<x<5)), so (x = 0) is not a local maximum or minimum.

Answer:

D.