First slide

Tangents and normals

Question

The number of points on the curve $y = x^{3} - 3 x$
at which tangent is parallel to x-axis

Moderate

A

4
B

3
C

2
D

1

Solution

$T h e g i v e n c u r v e i s y = x^{3} - 3 x$
$D i f f e r e n t i a t e b o t h s i d e s w i t h r e s p e c t t o x$
$\frac{d y}{d x} = 3 x^{2} - 3$
$3 x^{2} - 3 = 0, \sin c e t h e s l o p e o f t h e \tan g e n t p a r a l l e l t o x - a x i s i s z e r o .$
$x^{2} = 1 x = \pm 1$
$A t t w o p o i n t s \tan g e n t s t o t h e g i v e n c u r v e a r e p a r a l l e l t o x - a x i s$

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