Question

Find the point on the curve y = x² − 2x + 3, where the tangent is parallel to x-axis ?

Answer 1

The slope of the x-axis is 0.
Now, let (x₁, y₁) be the required point.
Here,

\[\text { Since, the point lies on the curve } . \]

\[\text { Hence,} y_1 = {x_1}^2 - 2 x_1 + 3 . . . \left( 1 \right)\]

\[\text { Now }, y = x^2 - 2x + 3\]

\[\frac{dy}{dx} = 2x - 2\]

\[\text { Slope of the tangent at }\left( x, y \right)= \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) =2 x_1 - 2\]

\[\text { Given }:\]

\[\text { Slope of the tangent at }\left( x_1 , y_1 \right)=\text { Slope of thex-axis }\]

\[ = 2 x_1 - 2 = 0\]

\[ \Rightarrow x_1 = 1\]

\[\text { and }\]

\[ y_1 = 1 - 2 + 3 = 2 [\text { From } (1)]\]

\[ \therefore \text { Required point }=\left( x_1 , y_1 \right)=\left( 1, 2 \right)\]