Question

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

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
• Since \(f^{\prime}(5)=0\), \(x = 5\) is a critical point. And because the function changes from decreasing (\(05\)), \(f(5)=-25\) is a local minimum.
• \(f^{\prime}(0) = 0\), but the function is decreasing on both sides of \(x = 0\) (from \(-5

Answer:

D.