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(0)=4,f(2)=0,f(4)= - 4 ) | ( f(0)=0,f(4)=0 ) |\n| :---: | :---: |\n| ( f(x)>0 ) on ( (-infty,0) ) and ( (4,infty) ); | ( f(x)<0 ) on ( (0,4) ) |\n| ( f(2)=0 ) | |\n| ( f(x)>0 ) on ( (2,infty) ) | ( f(x)<0 ) on ( (-infty,2) ) |\nchoose the correct graph of ( f ) below.\n○ a.\n○ b.\n○ c.\n○ d.

Answer

Explanation:

Step1: Analyze increasing and decreasing intervals

Since (f^{\prime}(x)>0) on ((-\infty,0)) and ((4,\infty)), the function (f(x)) is increasing on ((-\infty,0)) and ((4,\infty)). Since (f^{\prime}(x)<0) on ((0,4)), the function (f(x)) is decreasing on ((0,4)). So (x = 0) is a local maximum (because the function changes from increasing to decreasing at (x = 0)) and (x=4) is a local minimum (because the function changes from decreasing to increasing at (x = 4)).

Step2: Analyze concavity

Since (f^{\prime\prime}(x)<0) on ((-\infty,2)), the function (f(x)) is concave - down on ((-\infty,2)). Since (f^{\prime\prime}(x)>0) on ((2,\infty)), the function (f(x)) is concave - up on ((2,\infty)). And (x = 2) is an inflection point ((f^{\prime\prime}(2)=0) and concavity changes at (x = 2)).

Step3: Use function values

We know that (f(0)=4), (f(2)=0), (f(4)=-4)

Now let's check each option:

Option A:
- At (x = 0), it is a local maximum ((y = 4)), at (x=4) it is a local minimum ((y=-4)). The function is increasing on ((-\infty,0)) and ((4,\infty)), decreasing on ((0,4)). It is concave - down on ((-\infty,2)) and concave - up on ((2,\infty))
Option B: The function is decreasing on ((-\infty,0)) which is wrong as (f^{\prime}(x)>0) on ((-\infty,0))
Option C: The function is increasing on ((0,4)) which is wrong as (f^{\prime}(x)<0) on ((0,4))
Option D: The function is decreasing on ((4,\infty)) which is wrong as (f^{\prime}(x)>0) on ((4,\infty))