Limit Continuity and Differentiability 7 Question 21

21. Let $f: R \rightarrow R$ be any function. Define $g: R \rightarrow R$ by $g(x)=|f(x)|, \forall \quad x$. Then, $g$ is

$(2000,2 M)$

(a) onto if $f$ is onto

(b) one-one if $f$ is one-one

(c) continuous if $f$ is continuous

(d) differentiable if $f$ is differentiable

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Answer:

Correct Answer: 21. (a)

Solution:

  1. Given, $x e^{x y}=y+\sin ^{2} x$

On putting $x=0$, we get

$$ \begin{aligned} 0 \cdot e^{0} & =y+0 \\ y & =0 \end{aligned} $$

On differentiating Eq. (i) both sides w.r.t. $x$, we get

$$ 1 \cdot e^{x y}+x \cdot e^{x y} \quad x \cdot \frac{d y}{d x}+y=\frac{d y}{d x}+2 \sin x \cos x $$

On putting $x=0, y=0$, we get

$$ \begin{array}{rlrl} & & e^{0}+0(0+0) & =\frac{d y}{d x _{(0,0)}}+2 \sin 0 \cos 0 \\ \Rightarrow \quad \frac{d y}{d x} _{0,0} & =1 \end{array} $$



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