$f^{\prime \prime}(a), f^{\prime \prime}(b), f^{\prime \prime}(c)$ are

• A

  in A.P. only
• B

  in G.P. only
• Correct answer
  C

  in both A.P. and G.P.
• D

  neither in A.P. nor in G.P.

Hence, the correct answer is option: 'C'.