What is the maximum value of b for a given function with certain properties?

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Hi All,

I have a problem, and it is so confusing to me. I put it here in the hope of getting some helps to make it clear.

Thank you,

the problem is given below.

What is the greatest value of b for which any real valued function f that satisfies the following properties mus also satisfy f(1)<5

i) f is infinitively differentiable on the real numbers.
ii)f(0)=1, f'(0)=1, f"(0)=2
iii)|f'''(x)|<b for all x in [0,1]

A.1
B.2
C.6
D.12
E.24
 
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I would suggest looking at the Taylor series with the remainder in the form using the Mean Value Theorem, which ever one that is.
 
Oh, It is so wonderful suggestion !

So, it is solved. The correct answer is D (12)
 

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