Question

Найдите наименьшее значение функции: y=(x^2-39x+39)•e^2-x на отрезке [0;6]. ошаговый ход действий распишите пожалуйста.

Answer 1

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Answer 2

Finding the Minimum Value of a Function

To find the minimum value of the function \( y = (x^2 - 39x + 39) \cdot e^{2-x} \) on the interval \([0, 6]\), we can follow these steps:

1. Find the Critical Points: - Calculate the first derivative of the function to find the critical points where the minimum may occur.

2. Check Endpoints: - Evaluate the function at the endpoints of the interval to ensure the minimum value is not located there.

3. Determine the Minimum: - Compare the values obtained from the critical points and endpoints to find the minimum value of the function on the given interval.

Let's proceed with these steps to find the minimum value of the function.

Step 1: Find the Critical Points

The critical points of the function occur where the derivative is equal to zero or undefined. Let's calculate the first derivative of the function \( y = (x^2 - 39x + 39) \cdot e^{2-x} \) and find the critical points.

The first derivative of the function is: \[ y' = (2x - 39) \cdot e^{2-x} - (x^2 - 39x + 39) \cdot e^{2-x} \]

To find the critical points, we need to solve the equation \( y' = 0 \) for \( x \).

Step 2: Check Endpoints

We will also evaluate the function at the endpoints of the interval, which are \( x = 0 \) and \( x = 6 \), to ensure the minimum value is not located there.

Step 3: Determine the Minimum

After finding the critical points and evaluating the function at the endpoints, we will compare the values obtained to determine the minimum value of the function on the interval \([0, 6]\).

Let's proceed with the calculations to find the minimum value of the function.

Answer 3

Для нахождения наименьшего значения функции y=(x^2-39x+39)•e^2-x на отрезке [0;6] следует выполнить следующие шаги:

1. Найдем производную функции y по переменной x: y' = (2x-39)e^(2-x) - (x^2-39x+39)e^(2-x)

2. Найдем точки экстремума, приравняв производную к нулю и решив уравнение: (2x-39)e^(2-x) - (x^2-39x+39)e^(2-x) =