91Ó°ÊÓ

Establish the inequality \(e^{x}<\frac{1}{1-x}\) for all \(x<1\).

If \(f\) and \(g\) are convex on an interval \(I\), show that any linear combination \(\alpha f+\beta g\) is also convex provided \(\alpha\) and \(\beta\) are nonnegative.

Prove a second-order version of the mean value theorem. Let \(f\) be continuous on \([a, b]\) and twice differentiable on \((a, b)\). Then there exists \(c \in(a, b)\) such that $$ f(b)=f(a)+(b-a) f^{\prime}(a)+(b-a)^{2} \frac{f^{\prime \prime}(c)}{2 !} $$