More Problems on the Derivative

In these pages we discuss some problems related to the derivative concept. Most of the problems are challenging and not easy. We will give the answer to some of them The readers are encouraged to find the solutions for the rest. Send it and we will publish it. Enjoy.

Problem 1. Show that the function

$$f(x) = \left\{\begin{array}{lll}e^{-1/x^2}&\mbox{if $x \neq 0$}\\ 0 &\mbox{when $x=0$}\end{array}\right.$$

is differentiable at 0 at any order. Find f⁽ⁿ⁾(0), $n=1,2, \cdots$

Answer.

Problem 2. Let f(x) be a function defined on (-a,a), with a > 0. Assume that f(x) is continuous at 0 and

$$\lim_{x \rightarrow 0} \frac{f(x) - f(kx)}{x} = \alpha$$

where k satisfies 0 < k < 1.

(a)

Show that f(x) is differentiable at 0 and that $f'(0) = \displaystyle \frac{\alpha}{1-k}$.

(b)

Show that the conclusion obtained in (a) is still valid even when k > 1.

Answer.

Problem 3. For any $\alpha \geq 0$, set

$$f_{\alpha}(x) = \alpha x + x^2\sin\left(\frac{1}{x}\right)\;.$$

(a)

Show that $f_{\alpha}(x)$ is differentiable.

(b)

Is $f'_{\alpha}(x)$ continuous at 0?

(c)

Show that if $\alpha \leq 1$, then $f_{\alpha}(x)$ is not monotone on any open interval which contains 0.

Answer.

Problem 4. Let f(x) be a differentiable function. Assume that f'(x) is continuous. Define the sequence $\{x_n\}$ by

$$x_{n+1} = f(x_n),\;\;\mbox{and $x_0$ is given.}$$

Assume that $\{x_n\}$ is not almost constant. Show that if $\{x_n\}$ converges to $\alpha$, then

$$\vert f'(\alpha)\vert \leq 1\;.$$

Answer.

Problem 5. Consider the function f(x) = (x²-1)ⁿ, for $n \geq 1$. Show that

$$(x^2-1) f^{(n+2)}(x) + 2x f^{(n+1)}(x) -n(n+1)f^{(n)}(x) = 0\;.$$

Answer.

Problem 6. Consider the function $f(x) = \sqrt{x^2+1}$. Show that

$$(x^2+1) f^{(n+2)}(x) + (2n+1)x f^{(n+1)}(x) +(n+1)(n-1)f^{(n)}(x) = 0\;.$$

Then show that f⁽²ⁿ⁺¹⁾(0) = 0, for any $n \geq 0$.

Answer.

Problem 7. Let $f(x) = \ln(1 + \alpha x)$, where $\alpha > 0$. Let a and b be such that

$$-\frac{1}{\alpha} < a \leq b\;.$$

Show that there exists $c \in (a,b)$ such that

$$f(b) - f(a) = f'(c) (b-a)\;.$$

Find c.

Answer.

Problem 8. Show that e^x = 1-x has a unique solution. Then find that solution.

Answer.

Problem 9. Show that for any $x \in (0,\pi/2)$, we have

$$x + \frac{x^3}{3} < \tan(x)\;.$$

Answer.

Problem 10. Show that for any a and b such that 0 < a< b, we have

$$\frac{b-a}{b} < \ln\left(\frac{b}{a}\right) < \frac{b-a}{a}\;\cdot$$

Answer.

Problem 11. Let $f(x) = x-\cos(x)$.

(1)

Show that the equation $x - \cos(x) = 0$ has one solution x₀ in the interval $(\pi/6, \pi/4)$.

(2)

Show that there exists $c \in (x_0, \pi/4)$ such that

$$\frac{\pi - 2\sqrt{2}}{\pi - 4 x_0} = f'(c)\;.$$

Answer.

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