Question

Find the equations of all lines having slope 0 which are tangent to the curve .

Answer 1

The equation of the given curve is,

y= 1 x 2 −2x+3

The slope of the tangent at any point ( x,y ) is given as,

Slope= dy dx

The slope of the given curve is,

d( 1 x 2 −2x+3 ) dx = −( 2x−2 ) ( x 2 −2x+3 ) =− 2( x−1 ) ( x 2 −2x+3 )

It is given that slope of the tangent is 0, then

− 2( x−1 ) ( x 2 −2x+3 ) =0 −2( x−1 )=0 x=1

The coordinate of y when x=1 is,

y= 1 1−2+3 = 1 2

Hence, the given point is ( 1, 1 2 ).

Therefore, equation of tangent through a point ( x 1 , y 1 ) is given by,

y− y 1 =m( x− x 1 )

Here, m is the slope of the tangent.

Substitute ( 1, 1 2 ) for ( x 1 , y 1 ) and 0 for m.

y− 1 2 =0( x−1 ) y= 1 2

Thus, equation of the line with slope 0 and tangent to the curve is y= 1 2 .