Let f be any function defined on mathbfR and let it satisfy the condition : |f(x)-f(y)|le |(x-y)^2 | , forall(x,y)inmathbfR . If f(0)=1 , then :

Differentiation and its Application: Let f be any function defined on mathbfR and let it satisfy the… | JEE Main 2021

JEE Main 2021 (Online) 26th February Morning ShiftQuestion 1

Let $f$ be any function defined on $\mathbf{R}$ and let it satisfy the condition : $|f(x)-f(y)|\le\left|(x-y)^{2}\right|$, $\forall(x,y)\in\mathbf{R}$. If $f(0)=1$, then :

Differentiation and its Application

$f(x)>0,\ \forall x\in\mathbf{R}$✓ Correct answer$f(x)<0,\ \forall x\in\mathbf{R}$$f(x)=0,\ \forall x\in\mathbf{R}$$f(x)$ can take any value in $\mathbf{R}$

Solution

Concept. Dividing the condition by $|h|$ shows the difference quotient tends to $0$, so $f'\equiv0$. Formulas used. (1) $\left|\dfrac{f(x+h)-f(x)}{h}\right|\le|h|$Step 1 - Derivative. By (1), as $h\to0$ the quotient tends to $0$: $f'(x)=0$ for every $x$. Step 2 - Constant. So $f$ is constant, $f(x)=f(0)=1>0$. Answer: (A)

Differentiation and its Application - JEE Mathematics question (JEE Main 2021 (Online) 26th February Morning Shift)

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