Differentiation

Let $f$ be defined for all real $x$, and suppose that

for all real $x$ and $y$. Prove that $f$ is constant.

Suppose $f^{\prime }(x) > 0$ in $(a, b)$. Prove that $f$ is strictly increasing in $(a, b)$, and let $g$ be its inverse function. Prove that $g$ is differentiable, and that

Suppose $g$ is a real function on $\mathbb {R}^{1}$, with bounded derivative (say $\left|g^{\prime }\right| \leq M$). Fix $\varepsilon > 0$, and define $f(x) = x+\varepsilon g(x)$. Prove that $f$ is one-to-one if $\varepsilon$ is small enough. (A set of admissible values of $\varepsilon$ can be determined which depends only on $M$).

If

where $C_{0}, \ldots , C_{n}$ are real constants, prove that the equation

has at least one real root between 0 and 1.

Suppose $f$ is defined and differentiable for every $x > 0$, and $f^{\prime }(x) \rightarrow 0$ as $x \rightarrow +\infty$. Put $g(x) = f(x+1)-f(x)$. Prove that $g(x) \rightarrow 0$ as $x \rightarrow +\infty$.

Suppose

Suppose $f^{\prime }(x), g^{\prime }(x)$ exist, $g^{\prime }(x) \neq 0$, and $f(x) = g(x) = 0$. Prove that

(This holds also for complex functions).

Suppose $f^{\prime }$ is continuous on $[a, b]$ and $\varepsilon > 0$. Prove that there exists $\delta > 0$ such that

whenever $0 < \left|t-x\right| < \delta , a \leq x \leq b, a \leq t \leq b$. (This could be expressed by saying that $f$ is uniformly differentiable on $[a, b]$ if $f^{\prime }$ is continuous on $[a, b]$). Does this hold for vector-valued functions too?

Let $f$ be a continuous real function on $\mathbb {R}^{1}$, of which it is known that $f^{\prime }(x)$ exists for all $x \neq 0$ and that $f^{\prime }(x) \rightarrow 3$ as $x \rightarrow 0$. Does it follow that $f^{\prime }(0)$ exists?

Suppose $f$ and $g$ are complex differentiable functions on $(0,1)$, $f(x) \rightarrow 0, g(x) \rightarrow 0$, $f^{\prime }(x) \rightarrow A, g^{\prime }(x) \rightarrow B$ as $x \rightarrow 0$, where $A$ and $B$ are complex numbers, $B \neq 0$. Prove that

Compare with Example 5.18. Hint:

Apply Theorem 5.13 to the real and imaginary parts of $f(x) / x$ and $g(x) / x$.

Suppose $f$ is defined in a neighborhood of $x$, and suppose $f^{\prime \prime }(x)$ exists. Show that

Show by an example that the limit may exist even if $f^{\prime \prime }(x)$ does not. Hint: Use Theorem 5.13.

If $f(x) = \left|x\right|^{3}$, compute $f^{\prime }(x), f^{\prime \prime }(x)$ for all real $x$, and show that $f^{(3)}(0)$ does not exist.

Suppose $a$ and $c$ are real numbers, $c > 0$, and $f$ is defined on $[-1,1]$ by

Prove the following statements:

Let $f$ be a differentiable real function defined in $(a, b)$. Prove that $f$ is convex if and only if $f^{\prime }$ is monotonically increasing. Assume next that $f^{\prime \prime }(x)$ exists for every $x \in (a, b)$, and prove that $f$ is convex if and only if $f^{\prime \prime }(x) \geq 0$ for all $x \in (a, b)$.

Suppose $a \in \mathbb {R}^{1}$, $f$ is a twice-differentiable real function on $(a, \infty )$, and $M_{0}, M_{1}, M_{2}$ are the least upper bounds of $\left|f(x)\right|,\left|f^{\prime }(x)\right|,\left|f^{\prime \prime }(x)\right|$, respectively, on $(a, \infty )$. Prove that

Hint: If $h > 0$, Taylor's theorem shows that

for some $\xi \in (x, x+2 h)$. Hence

To show that $M_{1}^{2} = 4 M_{0} M_{2}$ can actually happen, take $a = -1$, define

and show that $M_{0} = 1, M_{1} = 4, M_{2} = 4$. Does $M_{1}^{2} \leq 4 M_{0} M_{2}$ hold for vector-valued functions too?

Suppose $f$ is twice-differentiable on $(0, \infty ), f^{\prime \prime }$ is bounded on $(0, \infty )$, and $f(x) \rightarrow 0$ as $x \rightarrow \infty$. Prove that $f^{\prime }(x) \rightarrow 0$ as $x \rightarrow \infty$.

Hint: Let $a \rightarrow \infty$ in Exercise 15.

Suppose $f$ is a real, three times differentiable function on $[-1,1]$, such that

Prove that $f^{(3)}(x) \geq 3$ for some $x \in (-1,1)$. Note that equality holds for $\frac{1}{2}\left(x^{3}+x^{2}\right)$. Hint: Use Theorem 5.15, with $\alpha = 0$ and $\beta = \pm 1$, to show that there exist $s \in (0,1)$ and $t \in (-1,0)$ such that

Suppose $f$ is a real function on $[a, b], n$ is a positive integer, and $f^{(n-1)}$ exists for every $t \in [a, b]$. Let $\alpha , \beta$, and $P$ be as in Taylor's theorem (5.15). Define

for $t \in [a, b], t \neq \beta$, differentiate

$n-1$ times at $t = \alpha$, and derive the following version of Taylor's theorem:

Suppose $f$ is defined in $(-1,1)$ and $f^{\prime }(0)$ exists. Suppose $-1 < \alpha_{n} < \beta_{n} < 1$, $\alpha_{n} \rightarrow 0$, and $\beta_{n} \rightarrow 0$ as $n \rightarrow \infty$. Define the difference quotients

Prove the following statements:

Formulate and prove an inequality which follows from Taylor's theorem and which remains valid for vector-valued functions.

Let $E$ be a closed subset of $\mathbb {R}^{1}$. We saw in Exercise 22, Chap. 4, that there is a real continuous function $f$ on $\mathbb {R}^{1}$ whose zero set is $E$. Is it possible, for each closed set $E$, to find such an $f$ which is differentiable on $\mathbb {R}^{1}$, or one which is $n$ times differentiable, or even one which has derivatives of all orders on $\mathbb {R}^{1}$?

Suppose $f$ is a real function on $(-\infty , \infty )$. Call $x$ a fixed point of $f$ if $f(x) = x$.

The function $f$ defined by

has three fixed points, say $\alpha , \beta , \gamma$, where

For arbitrarily chosen $x_{1}$, define $\left\{ x_{n}\right\}$ by setting $x_{n+1} = f\left(x_{n}\right)$.

The process described in part (c) of Exercise 22 can of course also be applied to functions that $\operatorname {map}(0, \infty )$ to $(0, \infty )$.

Fix some $\alpha > 1$, and put

Both $f$ and $g$ have $\sqrt{\alpha }$ as their only fixed point in $(0, \infty )$. Try to explain, on the basis of properties of $f$ and $g$, why the convergence in Exercise 16, Chap. 3, is so much more rapid than it is in Exercise 17. (Compare $f^{\prime }$ and $g^{\prime }$, draw the zig-zags suggested in Exercise 22).

Do the same when $0 < \alpha < 1$.

Suppose $f$ is twice differentiable on [a,b], $f(a) < 0, f(b) > 0, f^{\prime }(x) \geq \delta > 0$, and $0 \leq f^{\prime \prime }(x) \leq M$ for all $x \in [a, b]$. Let $\xi$ be the unique point in $(a, b)$ at which $f(\xi ) = 0$.

Complete the details in the following outline of Newton's method for computing $\xi$.

Suppose $f$ is differentiable on $[a, b], f(a) = 0$, and there is a real number $A$ such that $\left|f^{\prime }(x)\right| \leq A\left|f(x)\right|$ on $[a, b]$. Prove that $f(x) = 0$ for all $x \in [a, b]$. Hint: Fix $x_{0} \in [a, b]$, let

for $a \leq x \leq x_{0}$. For any such $x$,

Hence $M_{0} = 0$ if $A\left(x_{0}-a\right) < 1$. That is, $f = 0$ on [a, $x_{0}$ ]. Proceed.

Let $\phi$ be a real function defined on a rectangle $R$ in the plane, given by $a \leq x \leq b$, $\alpha \leq y \leq \beta$. A solution of the initial-value problem

is, by definition, a differentiable function $f$ on $[a, b]$ such that $f(a) = c, \alpha \leq f(x) \leq \beta$, and

Prove that such a problem has at most one solution if there is a constant $A$ such that

whenever $\left(x, y_{1}\right) \in R$ and $\left(x, y_{2}\right) \in R$. Hint: Apply Exercise 26 to the difference of two solutions. Note that this uniqueness theorem does not hold for the initial-value problem

which has two solutions: $f(x) = 0$ and $f(x) = x^{2} / 4$. Find all other solutions.

Formulate and prove an analogous uniqueness theorem for systems of differential equations of the form

Note that this can be rewritten in the form

where $\mathbf{y} = \left(y_{1}, \ldots , y_{k}\right)$ ranges over a $k$-cell, $\mathbf{\phi }$ is the mapping of a $(k+1)$-cell into the Euclidean $k$-space whose components are the functions $\phi_{1}, \ldots , \phi_{k}$, and $\mathbf{c}$ is the vector $\left(c_{1}, \ldots , c_{k}\right)$. Use Exercise 26, for vector-valued functions.

Specialize Exercise 28 by considering the system

where $f, g_{1}, \ldots , g_{k}$ are continuous real functions on $[a, b]$, and derive a uniqueness theorem for solutions of the equation

subject to initial conditions