JEE Main 2014 · 12 Apr · Shift 1 · Mathematics Q70

← JEE Main 2014 · 12 Apr · Shift 1

MathematicsSingle correct+4 / -1Question 70 of 90

Let $f,g:\mathbb R\to\mathbb R$ be two functions defined by
$f(x)=\begin{cases}x\sin\left(\dfrac1x\right),&x\ne0\\0,&x=0\end{cases},\ \text{and } g(x)=xf(x)$

Statement I : $f$ is a continuous function at $x=0$ .

Statement II : $g$ is a differentiable function at $x=0$ .

Both statements I and II are false.

Both statements I and II are true.

Statement I is true, statement II is false.

Statement I is false, statement II is true.

Answer and worked solution

Official answer: B

$|x\sin\frac1x|\le|x|\to0$ , so $f$ is continuous at 0.

$g(x)=x^2\sin\frac1x$ , and $\dfrac{g(h)-g(0)}h=h\sin\frac1h\to0$ , so $g$ is differentiable at 0. Both statements are true.

Topics:Differentiability Continuity

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JEE Main 2014 · 12 Apr · Shift 1, Mathematics question 70