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

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The discussion centers on determining the maximum value of b for a function f that meets specific criteria, including being infinitely differentiable and having defined values and derivatives at zero. The properties include f(0)=1, f'(0)=1, and f''(0)=2, along with the condition that the absolute value of the third derivative is less than b on the interval [0,1]. Participants suggest using the Taylor series and the Mean Value Theorem to analyze the problem. Ultimately, the consensus is that the correct maximum value of b is 12. This conclusion helps clarify the relationship between the function's properties and the constraints imposed on its derivatives.
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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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