Chapter 4

Suppose \(f: \mathbb{R} \rightarrow \mathbb{R}\) is continuously differentiable such that \(f^{\prime}(x)>0\) for all \(x .\) Show that \(f\) is invertible on the interval \(J=f(\mathbb{R}),\) the inverse is continuously differentiable, and \(\left(f^{-1}\right)^{\prime}(y)>0\) for all \(y \in f(\mathbb{R})\)

Compute the nth Taylor Polynomial at 0 for the exponential function.

Prove the product rule. Hint: Use \(f(x) g(x)-f(c) g(c)=f(x)(g(x)-g(c))+(f(x)-\) \(f(c)) g(c)\).

Suppose I, \(J\) are intervals and a monotone onto \(f: I \rightarrow J\) has an inverse \(g: J \rightarrow I\). Suppose you already know that both \(f\) and \(g\) are differentiable everywhere and \(f^{\prime}\) is never zero. Using chain rule but not Lemma 4.4.1 prove the formula \(g^{\prime}(y)=\frac{1}{f^{\prime}(g(y))} .\)

Suppose p is a polynomial of degree d. Given any \(x_{0} \in \mathbb{R}\), show that the \((d+1)\) th Taylor polynomial for \(p\) at \(x_{0}\) is equal to \(p\).

Prove the quotient rule. Hint: You can do this directly, but it may be easier to find the derivative of \(1 / x\) and then use the chain rule and the product rule.

Suppose \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a differentiable function such that \(f^{\prime}\) is a bounded function. Prove \(f\) is a Lipschitz continuous function.

Let \(f(x):=|x|^{3}\). Compute \(f^{\prime}(x)\) and \(f^{\prime \prime}(x)\) for all \(x\), but show that \(f^{(3)}(0)\) does not exist.

For \(n \in \mathbb{Z}\), prove that \(x^{n}\) is differentiable and find the derivative, unless, of course, \(n<0\) and \(x=0 .\) Hint: Use the product rule.

Suppose \(f:[a, b] \rightarrow \mathbb{R}\) is differentiable and \(c \in[a, b] .\) Then show there exists a sequence \(\left\\{x_{n}\right\\}\) converging to \(c, x_{n} \neq c\) for all \(n,\) such that $$ f^{\prime}(c)=\lim _{n \rightarrow \infty} f^{\prime}\left(x_{n}\right) $$ Do note this does not imply that \(f^{\prime}\) is continuous (why?).

Suppose \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a function such that \(|f(x)-f(y)| \leq|x-y|^{2}\) for all \(x\) and \(y .\) Show that \(f(x)=C\) for some constant \(C .\) Hint: Show that \(f\) is differentiable at all points and compute the derivative.

Define \(f: \mathbb{R} \rightarrow \mathbb{R}\) by $$f(x):=\left\\{\begin{array}{ll} x^{2} & \text { if } x \in \mathbb{Q} \\ 0 & \text { otherwise } \end{array}\right.$$ Prove that \(f\) is differentiable at \(0,\) but discontinuous at all points except \(0 .\)

Let \(f(x):=x+2 x^{2} \sin (1 / x)\) for \(x \neq 0\) and \(f(0):=0 .\) Show that \(f\) is differentiable at all \(x,\) that \(f^{\prime}(0)>0,\) but that \(f\) is not invertible on any open interval containing the origin.

Suppose \(f:[a, b] \rightarrow \mathbb{R}\) has \(n+1\) continuous derivatives and \(x_{0} \in(a, b) .\) Prove: \(f^{(k)}\left(x_{0}\right)=0\) for all \(k=0,1,2, \ldots, n\) if and only if \(g(x):=\frac{f(x)}{\left(x-x_{0}\right)^{n+1}}\) is continuous at \(x_{0}\).

Assume the inequality \(|x-\sin (x)| \leq x^{2} .\) Prove that sin is differentiable at \(0,\) and find the derivative at \(0 .\)

Suppose \(f:(a, b) \rightarrow \mathbb{R}\) is a differentiable function such that \(f^{\prime}(x) \neq 0\) for all \(x \in(a, b) .\) Suppose there exists a point \(c \in(a, b)\) such that \(f^{\prime}(c)>0 .\) Prove \(f^{\prime}(x)>0\) for all \(x \in(a, b) .\)

a) Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a continuously differentiable function and \(k>0\) be a number such that \(f^{\prime}(x) \geq k\) for all \(x \in \mathbb{R}\). Show \(f\) is one-to-one and onto, and has a continuously differentiable inverse \(f^{-1}: \mathbb{R} \rightarrow \mathbb{R}\). b) Find an example \(f: \mathbb{R} \rightarrow \mathbb{R}\) where \(f^{\prime}(x)>0\) for all \(x,\) but \(f\) is not onto.

Suppose \(a, b, c \in \mathbb{R}\) and \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable, \(f^{\prime \prime}(x)=a\) for all \(x, f^{\prime}(0)=b,\) and \(f(0)=c .\) Find \(f\) and prove that it is the unique differentiable function with this property.

Suppose \(f:(a, b) \rightarrow \mathbb{R}\) and \(g:(a, b) \rightarrow \mathbb{R}\) are differentiable functions such that \(f^{\prime}(x)=g^{\prime}(x)\) for all \(x \in(a, b)\), then show that there exists a constant \(C\) such that \(f(x)=g(x)+C\).

Suppose I, \(J\) are intervals and a monotone onto \(f: I \rightarrow J\) has an inverse \(g: J \rightarrow I .\) Suppose \(x \in I\) and \(y:=f(x) \in J,\) and that \(g\) is differentiable at y. Prove: a) If \(g^{\prime}(y) \neq 0,\) then \(f\) is differentiable at \(x\). b) If \(g^{\prime}(y)=0,\) then \(f\) is not differentiable at \(x\).

(Challenging): Show that a simple converse to Taylor's theorem does not hold. Find a \(f: \mathbb{R} \rightarrow \mathbb{R}\) with no second derivative at \(x=0\) such that \(|f(x)| \leq\left|x^{3}\right|,\) that is, \(f\) goes to zero at 0 fas \(x^{3},\) and while \(f^{\prime}(0)\) exists, \(f^{\prime \prime}(0)\) does not.

Let \(f: I \rightarrow \mathbb{R}\) be differentiable. Given \(n \in \mathbb{Z},\) define \(f^{n}\) be the function defined by \(f^{n}(x):=\) \((f(x))^{n} .\) If \(n<0,\) assume \(f(x) \neq 0 .\) Prove that \(\left(f^{n}\right)^{\prime}(x)=n(f(x))^{n-1} f^{\prime}(x)\)

Prove the following version of \(L\) 'Hôpital's rule. Suppose \(f:(a, b) \rightarrow \mathbb{R}\) and \(g:(a, b) \rightarrow \mathbb{R}\) are differentiable functions. Suppose that at \(c \in(a, b), f(c)=0, g(c)=0, g^{\prime}(x) \neq 0\) when \(x \neq c,\) and that the limit of \(f^{\prime}(x) / g^{\prime}(x)\) as \(x\) goes to \(c\) exists. Show that $$ \lim _{x \rightarrow c} \frac{f(x)}{g(x)}=\lim _{x \rightarrow c} \frac{f^{\prime}(x)}{g^{\prime}(x)} $$ Compare to Exercise 4.1.15.

Suppose \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a differentiable Lipschitz continuous function. Prove that \(f^{\prime}\) is \(a\) bounded function.

Let \(f:(a, b) \rightarrow \mathbb{R}\) be an unbounded differentiable function. Show \(f^{\prime}:(a, b) \rightarrow \mathbb{R}\) is unbounded.

Prove the \(n\) th derivative test. Suppose \(n \in \mathbb{N}, x_{0} \in(a, b),\) and \(f:(a, b) \rightarrow \mathbb{R}\) is n times continuously differentiable, with \(f^{(k)}\left(x_{0}\right)=0\) for \(k=1,2, \ldots, n-1,\) and \(f^{(n)}\left(x_{0}\right) \neq 0 .\) Prove: a) If \(n\) is odd, then \(f\) has neither a relative minimum, nor a maximum at \(x_{0}\). b) If \(n\) is even, then \(f\) has a strict relative minimum at \(x_{0}\) if \(f^{(n)}\left(x_{0}\right)>0\) and a strict relative maximum at \(x_{0}\) if \(f^{(n)}\left(x_{0}\right)<0\).

Let \(I_{1}, I_{2}\) be intervals. Let \(f: I_{1} \rightarrow I_{2}\) be a bijective function and \(g: I_{2} \rightarrow I_{1}\) be the inverse. Suppose that both \(f\) is differentiable at \(c \in I_{1}\) and \(f^{\prime}(c) \neq 0\) and \(g\) is differentiable at \(f(c) .\) Use the chain rule to find a formula for \(g^{\prime}(f(c))\) (in terms of \(\left.f^{\prime}(c)\right)\).

Prove the theorem Rolle actually proved in 1691: If \(f\) is a polynomial, \(f^{\prime}(a)=f^{\prime}(b)=0\) for some \(a

Prove the more general version of the second derivative test. Suppose \(f:(a, b) \rightarrow \mathbb{R}\) is differentiable and \(x_{0} \in(a, b)\) is such that, \(f^{\prime}\left(x_{0}\right)=0, f^{\prime \prime}\left(x_{0}\right)\) exists, and \(f^{\prime \prime}\left(x_{0}\right)>0 .\) Prove that \(f\) has a strict relative minimum at \(x_{0} .\) Hint: Consider the limit definition of \(f^{\prime \prime}\left(x_{0}\right) .\)

Suppose \(f: I \rightarrow \mathbb{R}\) is bounded and \(g: I \rightarrow \mathbb{R}\) is differentiable at \(c \in I\) and \(g(c)=g^{\prime}(c)=0 .\) Show that \(h(x):=f(x) g(x)\) is differentiable at c. Hint: You cannot apply the product rule.

Suppose \(a, b \in \mathbb{R}\) and \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable, \(f^{\prime}(x)=a\) for all \(x\), and \(f(0)=b\). Find \(f\) and prove that it is the unique differentiable function with this property.

Suppose \(f: I \rightarrow \mathbb{R}, g: I \rightarrow \mathbb{R},\) and \(h: I \rightarrow \mathbb{R},\) are functions. Suppose \(c \in I\) is such that \(f(c)=g(c)=h(c), g\) and h are differentiable at \(c,\) and \(g^{\prime}(c)=h^{\prime}(c) .\) Furthermore suppose \(h(x) \leq f(x) \leq g(x)\) for all \(x \in I\). Prove \(f\) is differentiable at \(c\) and \(f^{\prime}(c)=g^{\prime}(c)=h^{\prime}(c)\).

Suppose \(f:(-1,1) \rightarrow \mathbb{R}\) is a function such that \(f(x)=x h(x)\) for a bounded function \(h .\) a) Show that \(g(x):=(f(x))^{2}\) is differentiable at the origin and \(g^{\prime}(0)=0\). b) Find an example of a continuous function \(f:(-1,1) \rightarrow \mathbb{R}\) with \(f(0)=0,\) but such that \(g(x):=(f(x))^{2}\) is not differentiable at the origin.

Suppose \(f: I \rightarrow \mathbb{R}\) is differentiable at \(c \in I .\) Prove there exist numbers a and \(b\) with the property that for every \(\varepsilon>0,\) there is a \(\delta>0,\) such that \(|a+b(x-c)-f(x)| \leq \varepsilon|x-c|,\) whenever \(x \in I\) and \(|x-c|<\delta .\) In other words, show that there exists a function \(g: I \rightarrow \mathbb{R}\) such that \(\lim _{x \rightarrow c} g(x)=0\) and \(|a+b(x-c)-f(x)| \leq|x-c| g(x)\).

Prove the following simple version of \(L\) 'Hôpital's rule. Suppose \(f:(a, b) \rightarrow \mathbb{R}\) and \(g:(a, b) \rightarrow \mathbb{R}\) are differentiable functions whose derivatives \(f^{\prime}\) and \(g^{\prime}\) are continuous functions. Suppose that at \(c \in(a, b), f(c)=0, g(c)=0,\) and \(g^{\prime}(x) \neq 0\) for all \(x \in(a, b),\) and suppose that the limit of \(f^{\prime}(x) / g^{\prime}(x)\) as \(x\) goes to \(c\) exists. Show that $$\lim _{x \rightarrow c} \frac{f(x)}{g(x)}=\lim _{x \rightarrow c} \frac{f^{\prime}(x)}{g^{\prime}(x)}.$$