Calculus in functional equation

Calculus Level 4

Let $f: \mathbb R \to \mathbb R$ satisfy $| f(x+y) - f(x-y) - y | \leq y^2$ for all $(x,y) \in \mathbb R^2$ .

Determine which of the following can be a possible function of $f(x)$ .

Note: In the options, $c$ denote an arbitrary constant.

cos(x)+c sin(x)+c (x/2)+c (x^2)+c

1 solution

Rajdeep Brahma
Mar 25, 2017

Set x-y=z & 2y=h. Let t belong to R & t>0. If(z+h)-f(z)- $\frac{h}{2}$ I<= $\frac{h^2}{4}$ or I $\frac{f(z+h)-f(z)}{h}$ -1/2I<=h/4 Take IhI<=4t So I $\frac{f(z+h)-f(z)}{h}$ -1/2I<=t or
Id(f(z))/dz-1/2I<=t So,d(f(z))/dz=1/2 Integrating both sides we get f(z)=x/2+c

Eta ki MCQ chilo?. Then kora easy, nahole tough!

Md Zuhair - 4 years, 1 month ago

No,this was a subjective question of ISI ENTRANCE 2015.

rajdeep brahma - 4 years, 1 month ago

WTF! Its tough bro. Jodi mcq hoto then easy

Md Zuhair - 4 years, 1 month ago

Calculus in functional equation

1 solution

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