Question

Доведіть що функція у=(cosx/10)+x/8 зростає на множині дійсних чисел. ​

Answer 1

Пошаговое объяснение:

1) Область определения функции:

$D(y) = \R,$

т.е. функция определена на множ. действительных чисел.

2) Определим производную:

$y' = (\cos\tfrac{x}{10}+\tfrac{x}{8} )' = - \sin \tfrac{x}{10} {\cdot} ( \tfrac{x}{10})' + ( \tfrac{x}{8})' \\ = - \frac{\sin \tfrac{x}{10}}{10} + \frac{1}{8} = \frac{1}{8} - \frac{\sin \tfrac{x}{10}}{10}$

3) Сравним у'(х) с нулем:

$\small{ \frac{1}{8} { -} \frac{\sin \tfrac{x}{10}}{10}{ =} \frac{5}{40} { -}\frac{\sin \tfrac{x}{10}}{10}{ = } \frac{1}{40} { + }( \frac{4}{40} { - }\frac{\sin \tfrac{x}{10}}{10}) = } \\ = \frac{1}{40} + \frac{1}{10} (1 - {\sin \tfrac{x}{10}}) \\ \: 1 - {\sin \tfrac{x}{10}} \geqslant 0 \: < = >\frac{1}{10} (1 - {\sin \tfrac{x}{10}})\geqslant 0 \\ < = >\frac{1}{40} + \frac{1}{10} (1 - {\sin \tfrac{x}{10}})>0 \: \: \forall \: {x} \in \R = > \\ = > \: \forall \: {x} \in \R \: \: y'(x) > 0$

А то, что у'(х) > 0, означает, что функция у(х) возрастает на всей своей обл. определения, т.е.

у(х) возрастает на R

Что и требовалось доказать

Answer 2

Ответ:

Пошаговое объяснение:

Answer 3

Proof of Function Growth

To prove that the function u = (cosx/10) + x/8 increases for all real numbers, we can analyze the derivative of the function. If the derivative is positive for all real numbers, then the function is increasing.

The derivative of the function u with respect to x can be found using the sum and power rules of differentiation. The derivative of cosx is -sinx and the derivative of x/8 is 1/8. Therefore, the derivative of u is:

du/dx = (-sinx/10) + 1/8

Now, to show that the function is increasing for all real numbers, we need to demonstrate that the derivative du/dx is always greater than zero.

Let's analyze the sign of du/dx for all real numbers x.

Analysis of the Derivative

We know that -1 ≤ sinx ≤ 1 for all real numbers x. Therefore, -sinx/10 will always be between -1/10 and 1/10.

Adding 1/8 to -sinx/10 will always result in a positive value, as 1/8 is greater than 1/10.

This means that du/dx is always positive, indicating that the function u = (cosx/10) + x/8 is increasing for all real numbers x.

Conclusion

Therefore, we have shown that the function u = (cosx/10) + x/8 increases on the set of real numbers.

Note: The proof of the function's growth was based on the analysis of its derivative, which showed that the derivative is always positive for all real numbers.