# Understand the problem

Let be a twice differentiable function such thatShow that there exist such that for all .

##### Source of the problem

I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2019. Subjective Problem no. 4

##### Topic

calculus

##### Difficulty Level

8.5 out of 10

##### Suggested Book

# Problems In CALCULUS OF ONE VARIABLE-i.a maron

# Start with hints

Do you really need a hint? Try it first!

Do you know Taylor series expansion good then see…..

\(f(x+h) = f(x) + f'(x)h+f”(x)\frac{h^2}{2} + f”'(x)\frac{h^3}{3!}+\cdots\)

\(f(x-h) = f(x) – f'(x)h+f”(x)\frac{h^2}{2} – f”'(x)\frac{h^3}{3!}+\cdots\)

adding previous expression we get

\(\frac{f(x+h) – 2f(x) + f(x-h)}{h^2} = f”(x) + 2\frac{f””(x)}{4!}h^2+\cdots\)

taking the limit of the above equation as h goes to zero gives

\(\Rightarrow f”(x) = \lim_{h\to0} \frac{f(x+h) – 2f(x) + f(x-h)}{h^2} \,.\)

. Since is continuous, there exists such that The given identity becomes

Fix and differentiate the above identity with respect to and get

now we already get ,

If is twice differentiable at then

by rearranging our given problem we get that

Therefore , for some

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