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