Question

Провести полное исследование функции методами дифференциального исчисления и построить ее график. y=(x^2+1)/x

Answer 1

1. Находишь производную
2. Приравниваешь ее нулю
3. Решаешь полученное уравнение
4. Полученные решения - это абсциссы точек экстремума.
5. Подставляешь полученные значения в исходное уравнение и определяешь.

Answer 2

Function Analysis and Graph Plotting of y=(x^2+1)/x

To conduct a full analysis of the function y=(x^2+1)/x using methods of differential calculus and to plot its graph, we'll start by examining the function's properties and then proceed to differentiate it to find its critical points and inflection points. Finally, we'll plot the graph of the function.

Function Properties: The given function is y=(x^2+1)/x.

Differentiation: To find the derivative of the function, we'll use the quotient rule, which states that for two functions u(x) and v(x), the derivative of u(x)/v(x) is (v(x)u'(x) - u(x)v'(x))/[v(x)]^2.

Let's differentiate y=(x^2+1)/x step by step: 1. Let u(x) = x^2+1 and v(x) = x. 2. Then, u'(x) = 2x and v'(x) = 1. 3. Applying the quotient rule, the derivative of y with respect to x, y', is: y' = (x(x^2+1)' - (x^2+1)(x)') / x^2 y' = (x(2x) - (x^2+1)(1)) / x^2 y' = (2x^2 - x^2 - 1) / x^2 y' = (x^2 - 1) / x^2

Critical Points and Inflection Points: To find the critical points, we'll set the derivative equal to zero and solve for x: (x^2 - 1) / x^2 = 0 x^2 - 1 = 0 x^2 = 1 x = ±1

The critical points are x = -1 and x = 1.

To find the inflection points, we'll find the second derivative of the function and set it equal to zero.

Graph Plotting: Now, let's plot the graph of the function y=(x^2+1)/x using the critical points and inflection points we've found.

I'll now proceed to plot the graph of the function y=(x^2+1)/x.