First slide

Tangents and normals

Question

The points on the curve $y = x^{4} - 4 x^{3} + 4 x^{2} + 1$ at which the tangents are parallel to x-axis are

Moderate

A

$(0, 1), (1, 2), (2, 1)$
B

$(0, 1), (1, - 2), (2, - 1)$
C

$(0, - 1), (1, 2), (2, - 1)$
D

$(0, 1), (1, 2), (2, - 1)$

Solution

$y = x^{4} - 4 x^{3} + 4 x^{2} + 1$
Slope $= {(\frac{d y}{d x})}_{(x_{1}, y_{1})} = 4 x_{1}^{3} - 12 x_{1}^{2} + 8 x_{1}$

The points which tangent are parallel to x-axis are ${(\frac{d y}{d x})}_{(x_{1}, y_{1})} = 0$

$\Rightarrow 4 x_{1}^{3} - 12 x_{1}^{2} + 8 x_{1} = 0$

$\Rightarrow 4 x_{1} (x_{1}^{2} - 3 x_{1} + 2) = 0$

$\begin{matrix} ∴ x_{1} = 0, 1, 2 \\ y_{1} = 1, 2, 1 \end{matrix}} \Rightarrow (0, 1), (1, 2), (2, 1)$

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