Problem
ALG-B2-M05-P005 Tangent at one
#5
★★☆☆☆ Level 2 of 5
Prove for every real \(x\): \[x^2\ge2x-1.\]
Hint. Move everything to the left, or recall the tangent to \(x^2\) at \(1\).
We have \(x^2-2x+1=(x-1)^2\ge0\). This says exactly that the graph of the convex function \(x^2\) lies above its tangent at \(1\).
Tangents are the local version of convexity.