First slide

Tangents and normals

Question

$The curve y = x^{3} + x^{2} - x has two horizontal tangents. The distance between these two horizontal lines is$

Moderate

A

$\frac{13}{9}$
B

$\frac{11}{9}$
C

$\frac{22}{27}$
D

$\frac{32}{27}$

Solution

$\begin{array}{l} Given that f (x) = x^{3} + x^{2} - x = y \\ i f t h e \tan g e n t s a r e p a r a l l e l t o x - a x i s t h e n \frac{d y}{d x} = 0 \\ \Rightarrow 3 x^{2} + 2 x - 1 = 0 \\ ⇒ x = - 1, x = \frac{1}{3} \\ The tangents, at x = - 1, x = \frac{1}{3} are y = 1, y = \frac{- 5}{27} \end{array}$
$∴ Distance between horizontal lines = |y_{1} - y_{2}| = 1 + \frac{5}{27} = \frac{32}{27}$

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