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,
$$\begin{gathered} \mathrm{Since},\mathrm{the}\mathrm{point}\mathrm{lies}\mathrm{on}\mathrm{the}\mathrm{curve}. \\ \mathrm{Hence},y_1={x_1}^2-2x_1+3...\left(1\right) \\ \mathrm{Now},y=x^2-2x+3 \\ \frac{dy}{dx}=2x-2 \\ \text{Slope of the tangent at}\left(x,y\right)\text{=}{\left(\frac{dy}{dx}\right)}_{\left(x_1,y_1\right)}\text{=}2x_1-2 \\ \text{Given:} \\ \text{Slope of the tangent at}\left(x_1,y_1\right)\text{=Slope of the}\mathit{\text{x}}\text{-axis} \\ =2x_1-2=0 \\ \Rightarrow x_1=1 \\ \mathrm{and} \\ {y}_1=1-2+3=2[\text{From (1)]} \\ \therefore \text{Required point=}\left(x_1,y_1\right)\text{=}\left(1,2\right) \end{gathered}$$