EDB — 1TG

[1TG]Note:adapted from the written exam, April 9th, 2011.

Let \(f:[0,∞)→ℝ\) be a continuous function such that

• Fixed \(a{\lt}f(0)\), let \(M_ a\) be the set of \(m∈ℝ\) such that the line \(y=mx+a\) intersects the graph \(y=f(x)\) of the function \(f\) at least in one point: show that \(M_ a\) admits minimum \(\hat m=\hat m(a)\);

• show that \(\hat m\) depends continuously on \(a\), ¹

• and that \(\hat m(a)\) is monotonic strictly decreasing.

• If \(f\) is differentiable, show that the line \(y=\hat m(a) x+a\) is tangent to the graph at all points where it encounters it.

• Suppose further that \(f\) is of class \(C^ 2\) and that \(f''(x){\gt}0∀ x{\gt}0\) ² . Show that there is only one point \(x\) where the line \(y=\hat m(a) x+a\) meets the graph \(y=f(x)\); name it \(\hat x=\hat x(a)\);

• and show that the functions \(a↦ \hat x(a)\) and \(a↦ \hat m(a)\) are differentiable.