The curve is $y=x^2-2x+4,$ and a family of lines is given by $y=mx+1.$ Find all values of $m$

Question

The curve is $y=x^2-2x+4,$ and a family of lines is given by $y=mx+1.$ Find all values of $m$ for which the line is tangent to the curve. Find all values of $m$ for which the curve lies strictly above the line for every real $x$. Explain why the boundary values from (a) are excluded in (b).

Accepted Answer

(a) $m=-2\pm2\sqrt3$; (b) $-2-2\sqrt3<m<-2+2\sqrt3$; (c) Tangency gives equality at one point.

Quadratic Functions, Equations and Inequalities

Quadratic Functions, Equations and Inequalities