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Suppose we have a system of equations:
$x_{1} = f_{1} (x_{1}, . . ., x_{n}) x_{2} = f_{2} (x_{1}, . . ., x_{n}) . . . x_{n} = f_{n} (x_{1}, . . ., x_{n})$
where the $f_{i}$ are components of the gradient $\nabla f$ for a monotone increasing, concave function $f$ , so $f_{i} > 0$ and $\frac{\partial f_{i}}{\partial x_{i}} \leq 0$ .
Does the system of equations necessarily have a solution? And is it unique if it exists?

Cristofer Watson 2022-10-22

Suppose $f$ is a concave function on the interval $[a, b]$ , meaning
$λ f (x) + (1 - λ) f (y) \leq f (λ x + (1 - λ) y)$
for every $x, y \in [a, b]$ and every $λ \in [0, 1]$ . Prove that for any $p, q, r \in [a, b]$ with $q \geq r \geq 0$

Aarav Atkins 2022-10-22

How do I compute the next integral:
$\int \sqrt{\frac{x - 1}{x + 1}} d x ?$

Angel Kline 2022-10-22

Find all points of inflection of the function g(x)=x^2 e^{-x}

faois3nh 2022-10-21

How do you prove that the exponential function is continuous at x=0 and how do you prove it is continuous for all real x?

Aarav Atkins 2022-10-21

Compute the integral:
$\int_{- \infty}^{\infty} \frac{1}{y^{4} + 1} d y$

Cristofer Watson 2022-10-21

solve the below equation with respect to x
$0.6 \cdot \exp (\frac{- 40}{x}) + 0.4 \cdot \exp (\frac{10}{x}) = 1$

Ashleigh Clarke 2022-10-20

Need a find $\int_{.5}^{2} l n (4 + x^{2}) d x$ I understand that to find this value I need to find what the power series of $l n (4 + x^{2})$ and I know how to start this off since $\ln (1 - x) = \int \frac{1}{1 - x}$ which is $\frac{1}{1 - x}$ is the geometric series.

omgespit9q 2022-10-20

Prove $\int_{0}^{1} \frac{t^{2} - 1}{(t^{2} + 1) \log t} d t = 2 \log (\frac{2 Γ (\frac{5}{4})}{Γ (\frac{3}{4})})$

Evelyn Freeman 2022-10-20

How to show that $f^{- 1} (I)$ is convex if that is indeed the case?

rhenan5v 2022-10-20

Let $g (x) : R \to R^{+}$ be a measurable function. Suppose ${x : g (x) > 0} = (a, b)$ . If $g (x) - g (x - α) \geq g (y) - g (y - α), x < y, α \geq 0$
Can we say $g (.)$ is concave? How can show this?

Madilyn Quinn 2022-10-20

Solve the equation algebraically using logarithms. Round to the nearest hundredth
$7^{3 x + 9} = 12$

Brianna Schmidt 2022-10-20

Given the series definition of the exponential function, i.e. $\exp (x) = \sum_{k = 0}^{\infty} \frac{x^{k}}{k!}$ . Given that I have already proven that polynomials are continuous, does from this fact follow the continuity of the exponential function ?

Maribel Mcintyre 2022-10-20

$5 e^{2 x} = 6 + 29 e^{x}$
I would ususally multiply each side by a log value to get those exponents by themselfs, but that 6 is throwing me off. How do I get rid of it? Or do i just have to multiply it by log too?

Amira Serrano 2022-10-20

The indefinite integral $\int x^{2} \sqrt{1 - x} d x$

Rene Nicholson 2022-10-20

Evaluate :
$\int_{0}^{\frac{π}{2}} \ln^{2} (\cos^{2} x) d x$

Krish Logan 2022-10-20

What is the difference between this notation of the exponential function
$(1 + \frac{1}{n})^{n} \to e a s n \to \infty$
and this notation:
$lim_{n \to \infty} (1 + \frac{x}{n})^{n}$
Why is there a variable x in the second equation, and a 1 in the first equation?

Gisselle Hodges 2022-10-19

How to do integral $\int_{- \infty}^{\infty} x^{4} e^{- x^{2} / 2} d x$

podvelkaj8 2022-10-18

Equality of two trigonometric integrals on [0,1]
I need to show, that:
$\int_{0}^{1} \cos (x^{2}) d x = \frac{1}{2} \int_{0}^{1} \frac{\cos x}{\sqrt{x}} d x$
But frankly I cannot see way to solve it. The right-side integral is improper and as far I know both of them don't have the elementary antiderivatives.

Taniya Melton 2022-10-18

What is example of functions that grow faster than the exponential functions and/or factorial functions?

The majority of Calculus 1 questions and answers that you will find below should be sufficient for understanding the basic rules that are necessary for problem-solving purposes. Take a look at the original answers, seek math assignment help, and compare them to what you have for your assignment.

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