Answer to Question #192495 in Real Analysis for Nikhil

Question

Show that on the curve, y= 3x^2-7x+6, the chord joining the points whose abscissa are x=1 and x=2, parallel to the tangent at the point whose abscissa is x=3/2

Answer 1

Solution

Ordinates for two points of chord abscissa x₁=1 and x₂=2 are y₁=y(x₁)=2 and y₂= y(x₂)=4.

The slope of the chord is k=( y₂-y₁)/( x₂-x₁)=2.

The tangent at the any point of curve y(x) is the line with the slope equal to its derivative y’(x)=6x-7. At the abscissa x₀=3/2 the tangent is y’(3/2)=2.

So slopes of the chord and the tangent are equal. Therefore this lines are parallel.