Intereting Posts

Do eigenfunctions of elliptic operator form basis of $H^k(M)$?
Linear Transformations: Proving 1 dimensional subspace goes to 1 dimensional
Generalized Trigonometric Functions in terms of exponentials and roots of unity
How to show that $\Bbb Z/(xw-zy)$ is not a UFD
Formula to project a vector onto a plane
proving that $\frac{(n^2)!}{(n!)^n}$ is an integer
A polynomial with integer coefficient
Iterated Integrals and Riemann-Liouville (Fractional) Derivatives
Book recommendation for Putnam/Olympiads
Intriguing Indefinite Integral: $\int ( \frac{x^2-3x+1/3 }{x^3-x+1})^2 \mathrm{d}x$
Cubic diophantine equation
About the interior of the union of two sets
Gaining Mathematical Maturity
Companions to Rudin?
Why aren't all dense subsets of $\mathbb{R}$ uncountable?

The following is a homework problem:

Prove that between two roots of the function $f(x) = \exp(x) \sin(x) -1$ there must be at least one root of the function $\exp(x)\cos(x) +1$.

I think it is possible to prove it by simply computing the signs of the functions at $\frac{\pi}{2}k, \pi k, \frac{3\pi}{2}k, 2\pi k$. However, I’m looking for a more elegant way. It is intuitively clear that the roots of $f$ are ˋˋconverging” to the roots of $sin$ and the ones of $g$ to the ones of $cos$. Do you have any hint how to construct a nice argument based on this observation?

- Why does a circle enclose the largest area?
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- Law of iterated logarithms for BM
- Integral eigenvectors and eigenvalues
- Are there periodic functions without a smallest period?
- Defining a metric space

- What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity?
- how to solve binary form $ax^2+bxy+cy^2=m$, for integer and rational $ (x,y)$
- Prove that $\int_{0}^{\infty}{1\over x^4+x^2+1}dx=\int_{0}^{\infty}{1\over x^8+x^4+1}dx$
- Closed and exact.
- In more detail, why does L'Hospital's not apply here?
- Find the ratio of $\frac{\int_{0}^{1} \left(1-x^{50}\right)^{100} dx}{\int_{0}^{1} \left(1-x^{50}\right)^{101} dx}$
- Composition of Riemann integrable functions
- Difference between a Gradient and Tangent
- Showing that the product of vector magnitudes is larger than their dot product
- How to compute the following integral in $n$ variables?

The functions $g(x)=\sin(x)-\exp(-x)$ and $g'(x)=\cos(x)+\exp(-x)$ have the same roots as the given functions. Thus the problem reduces to the theorem of Rolle.

- Generating sequences using the linear congruential generator
- Will $2$ linear equations with $2$ unknowns always have a solution?
- Prove that $\|UVU^{-1}V^{-1}-I\|\leq 2\|U-I\|\|V-I\|$
- A non-UFD where we have different lengths of irreducible factorizations?
- Probability that two sets are disjoint? the same?
- List of matrix properties which are preserved after a change of basis
- On the grade of an ideal
- Space of bounded functions is reflexive if the domain is finite
- complex conjugates of holomorphic functions
- Generators of a finitely generated free module over a commutative ring
- What is operator calculus?
- Is the result true when the valuation is trivial and $\dim(X)=n$?
- $C^{1}$ function such that $f(0) = 0$, $\int_{0}^{1}f'(x)^{2}\, dx \leq 1$ and $\int_{0}^{1}f(x)\, dx = 1$
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- I cannot see why Ahlfors' statement is true (Extending a conformal map)