Differential Calculus

Outline a proof, starting from basic properties of the real numbers, of the following theorem: Let $f: [a,b] \rightarrow \mathbb {R}$ be a continuous function such that $f'(x) = 0$ for all $x \in (a,b)$. Then $f(b) = f(a)$.

Let $f(x) = x\log (1+x^{-1})$, $0 < x < \infty$.

Let $f(x)$, $0 \leq x < \infty$, be continuous, differentiable, with $f(0) = 0$, and that $f'(x)$ is an increasing function of $x$ for $x \geq 0$. Prove that

is an increasing function of $x$.

Let $y: \mathbb {R} \rightarrow \mathbb {R}$ be a $C^{\infty }$ function that satisfies the differential equation

for $x \in [0,L]$, where $L$ is a positive real number. Suppose that $y(0) = y(L) = 0$. Prove that $y \equiv 0$ on $[0,L]$.

Let $u(x)$, $0 \leq x \leq 1$, be a real valued $C^{2}$ function which satisfies the differential equation

Let $K$ be a real constant. Suppose that $y(t)$ is a positive differentiable function satisfying $y'(t) \leq Ky(t)$ for $t \geq 0$. Prove that $y(t) \leq e^{Kt}y(0)$ for $t \geq 0$.

Suppose $f$ is a differentiable function from the reals into the reals. Suppose $f'(x) > f(x)$ for all $x \in \mathbb {R}$, and $f(x_{0}) = 0$. Prove that $f(x) > 0$ for all $x > x_{0}$.

Suppose that $f$ is a twice differentiable real-valued function on $\mathbb {R}$ such that $f(0) = 0$, $f'(0) > 0$, and $f''(x) \geq f(x)$ for all $x \geq 0$. Prove that $f(x) > 0$ for all $x > 0$.

Show that the equation $ae^{x} = 1+x+x^{2}/2$, where $a$ is a positive constant, has exactly one real root.

Suppose $f: [0,1] \rightarrow \mathbb {R}$ is continuous with $f(0) = 0$, and for $0 < x < 1$, $f$ is differentiable and $0 \leq f'(x) \leq 2f(x)$. Prove that $f$ is identically 0.

Let $f: [0,1] \rightarrow \mathbb {R}$ be continuous function, with $f(0) = f(1) = 0$. Assume that $f''$ exists on $0 < x < 1$, with $f''+2f'+f \geq 0$. Show that $f(x) \leq 0$ for all $0 \leq x \leq 1$.

Let $v_{1}$ and $v_{2}$ be two real valued continuous functions on $\mathbb {R}$ such that $v_{1}(x) < v_{2}(x)$ for all $x \in \mathbb {R}$. Let $\varphi_{1}(t)$ and $\varphi_{2}(t)$ be, respectively, solutions of the differential equations

for $a < t < b$. If $\varphi_{1}(t_{0}) = \varphi_{2}(t_{0})$ for some $t_{0} \in (a,b)$, show that $\varphi_{1}(t) \leq \varphi_{2}(t)$ for all $t \in (t_{0}, b)$.

Prove or supply a counterexample: If $f$ and $g$ are $C^{1}$ real valued functions on $(0,1)$, if

if $g$ and $g'$ never vanish, and if

then

Let $f$ be a real-valued differentiable function on $(-1,1)$ such that $f(x)/x^{2}$ has a finite limit as $x \rightarrow 0$. Does it follow that $f''(0)$ exists? Give a proof or a counterexample.

Let $f$ be an infinitely differentiable function from $\mathbb {R}$ to $\mathbb {R}$. Suppose that, for some positive integer $n$,

Prove that $f^{(n+1)}(x) = 0$ for some $x$ in $(0,1)$.

Let $f$ be a positive function of class $C^{2}$ on $(0,\infty )$ such that $f' \leq 0$ and $f''$ is bounded. Prove that $\lim_{t \rightarrow \infty } f'(t) = 0$.

Let $f$ be a positive differentiable function on $(0,\infty )$. Prove that

exists (finitely) and is nonzero for each $x$.

Suppose that $f(x)$, $-\infty < x < \infty$, is a continuous real valued function, that $f'(x)$ exists for $x \neq 0$, and that $\lim_{x \rightarrow 0} f'(x)$ exists. Prove that $f'(0)$ exists.

For each real value of the parameter $t$, determine the number of real roots, counting multiplicities, of the cubic polynomial $p_{t}(x) = (1+t^{2})x^{3}-3t^{3}x+t^{4}$.

Let the real valued function $f$ be defined in an open interval about the point $a$ on the real line and be differentiable at $a$. Prove that if $(x_{n})$ is an increasing sequence and $(y_{n})$ is a decreasing sequence in the domain of $f$, and both sequences converge to $a$, then

Let $f$ be a continuous real valued function on $[0,1]$ such that, for each $x_{0} \in [0,1)$,

Prove that $f$ is nondecreasing.

Let $I$ be an open interval in $\mathbb {R}$ containing zero. Assume that $f'$ exists on a neighborhood of zero and $f''(0)$ exists. Show that

($o(x^{2})$ denotes a quantity such that $\frac{o(x^{2})}{x^{2}} \rightarrow 0$ as $x \rightarrow 0$).

Prove that the Taylor coefficients at the origin of the function

are rational numbers.

Give an example of a function $f: \mathbb {R} \rightarrow \mathbb {R}$ having all three of the following properties:

• $f(x) = 0$ for $x < 0$ and $x > 2$,

• $f'(1) = 1$,

• $f$ has derivatives of all orders.

Prove that if $n$ is a positive integer and $\alpha , \varepsilon$ are real numbers with $\varepsilon > 0$, then there is a real function $f$ with derivatives of all orders such that

1. $\left|f^{(k)}(x)\right| \leq \varepsilon$ for $k = 0, 1, \ldots , n-1$ and all $x \in \mathbb {R}$,

2. $f^{(k)}(0) = 0$ for $k = 0, 1, \ldots , n-1$,

3. $f^{(n)}(0) = \alpha$.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be continuously differentiable, periodic of period 1, and nonnegative. Show that

uniformly in $x$.

Let $I \subset \mathbb {R}$ be the open interval from 0 to 1. Let $f: I \rightarrow \mathbb {C}$ be $C^{1}$ (i.e., the real and imaginary parts are continuously differentiable). Suppose that $f(t) \rightarrow 0$, $f'(t) \rightarrow C \neq 0$ as $t \rightarrow 0+$. Show that the function $g(t) = \left|f(t)\right|$ is $C^{1}$ for sufficiently small $t > 0$ and that $\lim_{t \rightarrow 0+} g'(t)$ exists, and evaluate the limit.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be a $C^{\infty }$ function. Assume that $f(x)$ has a local minimum at $x = 0$. Prove there is a disc centered on the $y$ axis which lies above the graph of $f$ and touches the graph at $(0, f(0))$.