91Ó°ÊÓ

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 \(0 ?\)

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.

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.

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 \(f \in C^{(2)}(a, b)\). 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 \(c \in(a, b)\).

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 \(x>0\) $$ \sqrt{1+x}<1+\frac{x}{2} $$

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