Antiderivatives, working backwards.Sorry if my title is a bit confusing. I have the question:A function f defined for all real numbers satisfies: f ′ ( 1 ) = 1, f ( 0 ) = 4, and f ″ ( x ) = 12 x 2 − 12 x. Find the value of f(2).I left out some unimportant parts which I should be capable of doing myself after I understand this bit, my question is am I doing this correctly? f ′ ( x ) = 4 x 3 − 6 x 2 + C 4 ( 1 ) 3 − 6 ( 1 ) 2 + C = 1 4 − 6 + C = 1 C = 3 f ( x ) = x 4 − 2 x 3 + C f ( 0 ) = 0 4 − 2 ( 0 ) 3 + C = 4 C = 4 f ( 2 ) = 2 4 − 2 ( 4 ) 3 = C 16 − 2 ( 64 ) = C 16 − 128 = C − 112 = CIs this what I am being asked to do, or am I going about this the wrong way?

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Antiderivatives, working backwards.Sorry if my title is a bit confusing. I have the question:A function f defined for all real numbers satisfies:      f    ′    (  1  )  =  1,   f  (  0  )  =  4, and       f    ″    (  x  )  =  12      x    2    −  12  x. Find the value of f(2).I left out some unimportant parts which I should be capable of doing myself after I understand this bit, my question is am I doing this correctly?      f    ′    (  x  )  =  4      x    3    −  6      x    2    +  C  4  (  1      )    3    −  6  (  1      )    2    +  C  =  1  4  −  6  +  C  =  1  C  =  3  f  (  x  )  =      x    4    −  2      x    3    +  C  f  (  0  )  =      0    4    −  2  (  0      )    3    +  C  =  4  C  =  4  f  (  2  )  =      2    4    −  2  (  4      )    3    =  C  16  −  2  (  64  )  =  C  16  −  128  =  C  −  112  =  CIs this what I am being asked to do, or am I going about this the wrong way?

kubistiedt · Accepted Answer

Step 1Your first step (finding f′(x)) looks good, but from there you miss a few things. After finding the first C, you have       f    ′    (  x  )  =  4      x    3    −  6      x    2    +  3Step 2Thus, when we integrate, we have   f  (  x  )  =      x    4    −  2      x          3        +  3  x  +  C where here C is a different constant than the first one. Now we use   f  (  0  )  =  4 to find this C, after which you can plug in   x  =  2

Antiderivatives, working backwards. Sorry if my title is a bit confusing. I have the question: A

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