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How to integrate $\int \log (e^{x} + e^{- x}) d x$

Cortez Clarke 2022-11-03

Example of strictly concave functions?

Alfredo Cooley 2022-11-03

Let $f : R^{n} \times R^{m} \to R$ be a continuous function. Suppose that there exists an $a_{0} \in R^{m}$ such that $f (x, a_{0})$ is strictly concave in $x$ . Is it true that there exists a ball $A$ around $a_{0}$ such that $f (x, a)$ is concave in $f (x, a)$ for $a \in A$ ?

unabuenanuevasld 2022-11-03

Prove: $g (t x) = t \cdot g (x), \forall t \geq 0$

trumansoftjf0 2022-11-03

I have a function $f (x)$ which is a concave function, ( $x$ is a two element vector whose entities can take any value from the real line). I think since $\log (x)$ is a concave function therefore $\log (f (x))$ should be a concave function too. Reasoning right or wrong?

Filloltarninsv9p 2022-11-03

Find the derivative of this function using logarithmic differentiation:
$y = \frac{(x + 3)^{4} (2 x^{2} + 5 x)^{3}}{\sqrt{4 x - 3}}$

tramolatzqvg 2022-11-02

However, it is not clear to me that the same is true for quasi-concave functions $f$ ...
$f \mapsto \int_{x \in X} f (x) d μ$
quasi-concave on $X$ is $f$ is quasi-concave?

Emmanuel Giles 2022-11-02

Write the equation in exponential form. Assume that all constants are positive and not equal to 1
$\log_{z} (y) = c$

django0a6 2022-11-02

Find the derivative of the function below and simplify the result.
$g (x) = \ln (\frac{\sqrt{4 + x^{2}}}{x})$

MISA6zh 2022-11-02

Find the derivative of the expression
$e^{x} \cos x$

pin1ta4r3k7b 2022-11-02

Solve this one using exponential function
$\int \frac{2^{\frac{1}{x}}}{x^{2}} d x$

Demarion Ortega 2022-11-02

We have following function $f (x, y) = min (y (1 - x), x y a)$ where $x \geq 0, y \geq 0, a \geq 0$ . Only x and y are the optimization variables. I want to ask is it a concave function of $(x, y)$ ?

paratusojitos0yx 2022-11-02

Let $f (x)$ be strictly increasing and strictly convex, and $g (x)$ be strictly increasing and concave, then is also true that $h (x) = f (x) g (x)$ is convex?

Hayley Mcclain 2022-11-02

Is a function concave if and only if $f (x) / x$ decreasing?

Frances Pham 2022-11-02

Let $f : [0, 1] \to [0, 1]$ be a strictly increasing continuous concave function with $f (0) = 0$ and $f (1) = 1$ . Let $g$ be the inverse of $f$ . Then $g$ is strictly increasing and convex.
It seems that the function $h (x) = f (\frac{1}{2} g (x))$ is always convex. Is this true? If yes, why?

Kayley Dickson 2022-11-02

Why do convex and concave function have their own name like that not the opposite?

MISA6zh 2022-11-02

Solve for x.
$11^{6 x} = 8^{x - 5}$

Jadon Johnson 2022-11-02

Proving that $\exp (- x^{2} - y^{2})$ is a strictly quasi-concave function

gasavasiv 2022-11-01

I have a topological field K that admits a non-trivial continuous exponential function E, must every non-trivial continuous exponential function E′ on K be of the form $E^{'} (x) = E (r σ (x))$ for some $r \in K$ * and $σ \in A u t (K / Q)$ ?
If not, can you explain for which fields other than $R$ is this condition met?

erwachsenc6 2022-11-01

Let $f (A) = (det (A))^{\frac{1}{n}}$ . And assume domain of $f$ is space of positive semi definite symmetric $n \times n$ matrices with real entries. Show that $f$ is concave:
$f ((1 - t) A + t B)) \geq (1 - t) f (A) + t f (B)$
for $t \in [0, 1]$ .

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