Chapter 6

Suppose \(D \subset \mathbb{R}, a\) is an interior point of \(D, f: D \rightarrow \mathbb{R},\) and \(f^{\prime \prime}(a)\) exists. Show that $$ \lim _{h \rightarrow 0} \frac{f(a+h)+f(a-h)-2 f(a)}{h^{2}}=f^{\prime \prime}(a) $$ Find an example to illustrate that this limit may exist even if \(f^{\prime \prime}(a)\) does not exist.

Use l'Hôpital's rule to compute $$ \lim _{x \rightarrow 0^{+}} \frac{\sqrt{1+x}-1}{x} $$

Define \(g:(-1,1) \rightarrow \mathbb{R}\) by $$ g(x)=\left\\{\begin{aligned} -1, & \text { if }-1

Suppose \(f\) is differentiable on \((a, b)\) and \(f^{\prime}(x) \neq 0\) for all \(x \in(a, b) .\) Show that for any \(x, y \in(a, b), f(x) \neq f(y)\)

Show that if \(c \in \mathbb{R}\) and \(f(x)=c\) for all \(x \in \mathbb{R},\) then \(f^{\prime}(x)=0\) for all \(x \in \mathbb{R}\).

Show that if \(f: \mathbb{R} \rightarrow \mathbb{R}\) is linear, then there exists \(m \in \mathbb{R}\) such that \(f(x)=m x\) for all \(x \in \mathbb{R}\).

Use the 5 th order Taylor polynomial for \(f(x)=\sqrt{x}\) at 1 to estimate \(\sqrt{1.2}\). Is this an underestimate or an overestimate? Find an upper bound for the largest amount by which the estimate and \(\sqrt{1.2}\) differ.

Suppose \(a, b \in \mathbb{R}, f\) and \(g\) are differentiable on \((a, b), g^{\prime}(x) \neq 0\) for all \(x \in(a, b),\) and $$ \lim _{x \rightarrow b^{-}} \frac{f^{\prime}(x)}{g^{\prime}(x)}=\lambda $$ Show that if \(\lim _{x \rightarrow b^{-}} f(x)=0\) and \(\lim _{x \rightarrow b^{-}} g(x)=0,\) then $$ \lim _{x \rightarrow b^{-}} \frac{f(x)}{g(x)}=\lambda $$

Suppose \(f\) is differentiable on an open interval \(I\). Show that \(f^{\prime}\) cannot have any simple discontinuities in \(I\).

Define \(f:[0,+\infty) \rightarrow[0,+\infty)\) by \(f(x)=\sqrt{x}\). Show that \(f^{\prime}:(0,+\infty) \rightarrow(0,+\infty)\) is given by $$ f^{\prime}(x)=\frac{1}{2 \sqrt{x}} $$

Find the 3 rd order Taylor polynomial for \(f(x)=\sqrt[3]{1+x}\) at 0 and use it to estimate \(\sqrt[3]{1.1}\). Is this an underestimate or an overestimate? Find an upper bound for the largest amount by which the estimate and \(\sqrt[3]{1.1}\) differ.

Let \(f(x)\) be a third degree polynomial. Show that the equation \(f(x)=0\) as at least one, but no more than three, solutions.

Define \(f: \mathbb{R} \rightarrow \mathbb{R}\) by $$ f(x)=\left\\{\begin{array}{ll} x, & \text { if } x<0 \\ x^{2}, & \text { if } x \geq 0 \end{array}\right. $$ Is \(f\) differentiable at \(0 ?\)

Suppose \(f \in C^{(2)}(a, b)\). Use Taylor's theorem to show that $$ \lim _{h \rightarrow 0} \frac{f(c+h)+f(c-h)-2 f(c)}{h^{2}}=f^{\prime \prime}(c) $$ for any \(c \in(a, b)\).

Prove the Mean Value Theorem using Rolle's theorem and the function $$ k(t)=f(t)-\left(\left(\frac{f(b)-f(a)}{b-a}\right)(t-a)+f(a)\right) . $$ Give a geometric interpretation for \(k\) and compare it with the function \(h\) used in the proof of the generalized mean value theorem.

Define \(f: \mathbb{R} \rightarrow \mathbb{R}\) by $$ f(x)=\left\\{\begin{array}{ll} x^{2}, & \text { if } x<0 \\ x^{3}, & \text { if } x \geq 0 \end{array}\right. $$ Is \(f\) differentiable at \(0 ?\)

Suppose \(f \in C^{(1)}(a, b), c \in(a, b), f^{\prime}(c)=0,\) and \(f^{\prime \prime}\) exists on \((a, b)\) and is continuous at \(c .\) Show that \(f\) has a local maximum at \(c\) if \(f^{\prime \prime}(c)<0\) and a local minimum at \(c\) if \(f^{\prime \prime}(c)>0\).

Let \(a, b \in \mathbb{R}\). Suppose \(f\) is continuous on \([a, b]\), differentiable on \((a, b),\) and \(\left|f^{\prime}(x)\right| \leq M\) for all \(x \in(a, b)\). Show that $$ |f(b)-f(a)| \leq M|b-a| $$

Show that for all \(x>0\) $$ \sqrt{1+x}<1+\frac{x}{2} $$

Suppose \(I\) is an open interval, \(f: I \rightarrow \mathbb{R}\), and \(f^{\prime}(x)=0\) for all \(x \in I\). Show that there exists \(\alpha \in \mathbb{R}\) such that \(f(x)=\alpha\) for all \(x \in I\).

Given \(n \in \mathbb{Z}^{+}\) and \(f(x)=x^{n}\), use induction and the product rule to show that \(f^{\prime}(x)=n x^{n-1}\).

Suppose \(I\) is an open interval, \(f: I \rightarrow \mathbb{R}, g: I \rightarrow \mathbb{R},\) and \(f^{\prime}(x)=g^{\prime}(x)\) for all \(x \in I .\) Show that there exists \(\alpha \in \mathbb{R}\) such that $$ g(x)=f(x)+\alpha $$ for all \(x \in I\).

Show that for any integer \(n \neq 0,\) if \(f(x)=x^{n},\) then \(f^{\prime}(x)=\) \(n x^{n-1} .\)