Question

Find points at which the tangent to the curve y = x 3 − 3 x 2 − 9 x + 7 is parallel to the x -axis.

Answer 1

The equation of the curve is given by,

y= x 3 −3 x 2 −9x+7

The slope of the tangent of the curve is,

dy dx = d( x 3 −3 x 2 −9x+7 ) dx =3 x 2 −6x−9

For the tangent to be parallel to the x axis, the slope of the tangent should be 0.

dy dx =0 3 x 2 −6x−9=0 ( x−3 )( x+1 )=0

The solutions to the above equation are x=3 or −1.

The value of y when x=3 is given as,

( x 3 −3 x 2 −9x+7 ) x=3 =( 3 3 −3 ( 3 ) 2 −3( 3 )+7 ) =−20

The value of y when x=−1 is given as,

( x 3 −3 x 2 −9x+7 ) x=−1 =[ ( −1 ) 3 −3 ( −1 ) 2 −9( −1 )+7 ] =12

Thus, the coordinates at which the tangent is parallel to the curve are ( 3,20 ) and ( −1,12 ).