0
votes
1answer
35 views

Proving that $\lim\limits_{z\to z_0}\frac{f(z)}{g(z)}=\lim\limits_{z\to z_0}\frac{f^\prime(z)}{g^\prime(z)}$

Let $f,g$ both analythic in neighbourhood of $z_0$ and they both have zero of multiplicity $n$ in $z_0$. Prove that $\lim\limits_{z\to z_0}\frac{f(z)}{g(z)}=\lim\limits_{z\to ...
3
votes
0answers
55 views

Roots of derivative of q-expontial function

Let the q-deformation of the exponential function be defined by $$ e_q(z)=\sum_{n=0}^\infty{\frac{z^n}{[n]_q!}}. $$ Eq. (1.8) of this paper provides the product representation $$ ...
1
vote
1answer
66 views

Describe the graph of f if the graph of its integral its given

Describe the graph of $f$ if the graph of its integral $g(t) = \int_{0}^{t} f(s) ds $ is: graphic of g graphic of f I analyze the derivative and the sign of the derivative and try to find ...
1
vote
1answer
32 views

description of the function whose graph corresponds to Figure

Consider f be a real continuous function , $f(0) = 0$ , and whose graph has the form shown in the figure: a) How can a give description of the function whose graph corresponds to Figure. b) Sketch ...
0
votes
2answers
58 views

n'th and (n-1)'th derivative of $\frac{1}{x^n - a}$

I have a function of the form $f(x) = \frac{1}{x^n - a}$, and I need to programmatically find the n'th and (n-1)'th derivative of the function. Since the function has this specific form and that the ...
2
votes
0answers
41 views

Showing that the n first derivatives of (x²-1)^n have at least r roots (for the r-th derivative)?

I have f(x) = (x²-1)^n. I want to show that, for r = 0,1,2,...,n, the r-th derivative is a polynomial (that's easy to show) that has no fewer than r distinct roots in (-1,1). I guess I need to use ...
0
votes
1answer
101 views

If f is n-times differentiable, and $f^n$ is never 0, then f has at most n zeros in R

Let $n \ge 0$, let $f:\mathbb{R} \rightarrow \mathbb{R}$ be n-times diff erentiable on $\mathbb{R}$, and assume that $f^{(n)}(x) \neq 0$ for all $x \in \mathbb{R}$. Show that $f$ has at most $n$ zeros ...
3
votes
3answers
3k views

Prove using Rolle's Theorem that an equation has exactly one real solution.

So the question is; Prove that the equation $x^7+x^5+x^3+1=0$ has exactly one real solution. You should use Rolle’s Theorem at some point in the proof. And I have, Since $f(x) = x^7+x^5+x^3+1$ ...
3
votes
2answers
79 views

Number of real roots of the equation $2^x = 1+x^2$

Find the number of real roots of the equation $2^x = 1+x^2$ My try: Let we take $f(x) = 2^x-1-x^2$. Now for Drawing Graph of given function, we use Derivative Test. $f'(x) = 2^x \cdot \ln ...
1
vote
1answer
166 views

finding value of constant such that function has distinct root

we have the function cubic function $$ x^3 -12x +k =0 $$ it has distinct root in $$ [0,2{]} $$ that task given to us is to find the the value of k satisfying the above conditions I proceeded ...
7
votes
3answers
244 views

Is the complex derivative “speed”?

The first thing I was told about the real derivative is that it's "how fast the function is growing" at a given point. This interpretation wasn't addressed in my complex analysis classes. Can the ...
0
votes
1answer
69 views

Critical numbers of $x^{-5}\log x$

On a homework problem, I got the wrong answer and figured out what to put in for it to be marked correct (online homework), but I am unsure why it is right. The problem is to find the critical ...
0
votes
2answers
620 views

How to prove that r is a double root if and only if it is a root of a polynomial and of its derivative.

I don't know how to start the question. The title is self explanatory. How to approach and make a formal proof?
0
votes
1answer
74 views

First derivative

What would be the further steps for the case like this: I am finding the first derivative of a function: $f(x) = \ln(1+x^2)$ So the procedure would then be: $f'(x) = \frac{2x}{1+x^2}$. $f'(x) ...