91Ó°ÊÓ

For each of the following statements, determine whether it is true or false and justify your answer. a. If the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is continuous at \(x_{0},\) then it is differentiable at \(x_{0}\). b. If the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable at \(x_{0},\) then it is continuous at \(x_{0}\). c. The function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable if the function \(f^{2}: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable.

Define \(f(x)=1 /(1+x)\) for \(x\) in \(I \equiv(0,1)\). Show that \(f: I \rightarrow \mathbb{R}\) is strictly decreasing and differentiable and that \(f(I)=(1 / 2,1) \equiv J .\) Show that \(f^{-1}(y)=\) \((1-y) / y\) for \(y\) in \(J .\) Calculate the derivative of the inverse directly and then check that this calculation agrees with formula (4.6).

Prove that the following equation has exactly one solution: $$ x^{5}+5 x+1=0, \quad-1

Let \(I\) be a neighborhood of \(x_{0}\) and suppose that the function \(f: I \rightarrow \mathbb{R}\) has two continuous derivatives. Prove that $$ \lim _{h \rightarrow 0} \frac{f\left(x_{0}+h\right)-2 f\left(x_{0}\right)+f\left(x_{0}-h\right)}{h^{2}}=f^{\prime \prime}\left(x_{0}\right) $$

Suppose that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable and that \(\left\\{x_{n}\right\\}\) is a strictly increasing bounded sequence with \(f\left(x_{n}\right) \leq f\left(x_{n+1}\right)\) for all \(n\) in \(\mathbb{N} .\) Prove that there is a number \(x_{0}\) at which \(f^{\prime}\left(x_{0}\right) \geq 0 .\) (Hint: Apply the Monotone Convergence Theorem.)

Suppose that the function \(g: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable at \(x=0 .\) Also, suppose that for each natural number \(n, g(1 / n)=0 .\) Prove that \(g(0)=0\) and \(g^{\prime}(0)=0 .\)

Suppose that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable, that \(f^{\prime}: \mathbb{R} \rightarrow \mathbb{R}\) is continuous at \(0,\) and that \(f^{\prime}(0)>0 .\) Prove that there is an open interval \(I\) containing 0 such that \(f: I \rightarrow \mathbb{R}\) is strictly monotonic.