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Assertion (A): The tangent to the curve $y=x^{3}-x^{2}-x+2$ at $(1,1)$ is parallel to the $x$ -axis. Reason $(\mathbf{R}):$ The slope of the tangent to the curve at $(1,1)$ is zero.
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Both A and Rare true and $\mathrm{R}$ is the correct explanation
of A
of A
Given equation of the curve is $y=x^{3}-x^{2}-x+2$
$\frac{d y}{d x}=3 x^{2}-2 x-1$
$\Rightarrow\left(\frac{d y}{d x}\right)_{(1,1)}=3-2-1=0$
Since, $\frac{d y}{d x}$ at $(1,1)$ is 0 $\therefore$ slope of the tangent $=0$ Hence, The equation of tangent at $(1,1)$ is $y-1=0(x-1) \Rightarrow y=1$
ie, parallel to $x$ -axis. Both A and R true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$
$\frac{d y}{d x}=3 x^{2}-2 x-1$
$\Rightarrow\left(\frac{d y}{d x}\right)_{(1,1)}=3-2-1=0$
Since, $\frac{d y}{d x}$ at $(1,1)$ is 0 $\therefore$ slope of the tangent $=0$ Hence, The equation of tangent at $(1,1)$ is $y-1=0(x-1) \Rightarrow y=1$
ie, parallel to $x$ -axis. Both A and R true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$
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