Let \(I\) be an interval containing more than one point and \(f: I \rightarrow
\mathbb{R}\) be any function.
(i) Assume that \(f\) is differentiable. If \(f^{\prime}\) is nonnegative on \(I\)
and \(f^{\prime}\) vanishes at only a finite number of points on any bounded
subinterval of \(I\), then show that \(f\) is strictly increasing on \(I\).
(ii) Assume that \(f\) is twice differentiable. If \(f^{\prime \prime}\) is
nonnegative on \(I\) and \(f^{\prime \prime}\) vanishes at only a finite number of
points on any bounded subinterval of \(I\), then show that \(f\) is strictly
convex on \(I\).
(iii) Consider \(f: \mathbb{R} \rightarrow \mathbb{R}\) given by \(f(x)=(x-2
n)^{3}+2 n\), where \(n \in \mathbb{Z}\) is such that \(x \in[2 n-1,2 n+1) .\) Show
that \(f\) is differentiable on \(\mathbb{R}\) and \(f^{\prime \prime}\) exists on
\((2 n-1,2 n+1)\), but \(f_{+}^{\prime \prime}(2 n+1)=6\), whereas \(f_{-}^{\prime
\prime}(2 n+1)=-6\)
for each \(n \in \mathbb{N} .\) Also show that \(f\) is strictly increasing on
\(\mathbb{R}\) although \(f^{\prime}(2 n)=0\) for each \(n \in \mathbb{N}\).
(Compare (i) above and Exercise 12 in the list of Revision Exercises at the
end of Chapter 7.)
(iv) Consider \(g: \mathbb{R} \rightarrow \mathbb{R}\) given by \(g(x)=(x-2
n)^{4}+8 n x\), where \(n \in \mathbb{Z}\) is such that \(x \in[2 n-1,2 n+1) .\)
Show that \(g\) is twice differentiable on \(\mathbb{R}\) and \(g^{\prime \prime
\prime}\) exists on \((2 n-1,2 n+1)\), but \(g_{+}^{\prime \prime \prime}(2
n+1)=24\), whereas
\(g_{-}^{\prime \prime \prime}(2 n+1)=-24\) for each \(n \in \mathbb{N} .\) Also
show that \(g\) is strictly convex on \(\mathbb{R}\) although \(g^{\prime \prime}(2
n)=0\) for each \(n \in \mathbb{N}\). (Compare (ii) above and Exercise 13 in the
list of Revision Exercises at the end of Chapter \(7 .\) )
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine 
