Question

y=x^4-4x^3-8x^2+3 Знайдіть найбільше і найменше значення ф-ії на проміжку [-2; 0] Очень нужно! Ребяятааа, спасите меня от строгого завуча!

Answer 1

y'=4x^3-12x^2-16x=0

x=0 x^2-3x-4=0 x1=4 x2=-1;

отрезку принадлежит корень x=-1

y(0)=3

y(-2)=16+32-32+3=19

y(-1)=1+4-8+3=0

минимум y(-1)=0

максимум y(-2)=19

Answer 2

Finding the Maximum and Minimum Values of a Function

To find the maximum and minimum values of the function f(x) = x^4 - 4x^3 - 8x^2 + 3 on the interval \([-2, 0]\), we can follow these steps:

1. Find the Critical Points: - To find the critical points, we need to find the derivative of the function and solve for \(f'(x) = 0\).

2. Test for Maximum and Minimum: - Once we have the critical points, we can test the intervals around these points to determine whether they correspond to maximum or minimum values.

3. Check the Endpoints: - Finally, we need to check the function's values at the endpoints of the given interval.

Let's start by finding the critical points of the function.

Finding the Derivative

The derivative of the function \(f(x) = x^4 - 4x^3 - 8x^2 + 3\) is given by:

\[f'(x) = 4x^3 - 12x^2 - 16x\]

Finding the Critical Points

To find the critical points, we need to solve for \(f'(x) = 0\):

\[4x^3 - 12x^2 - 16x = 0\]

Solving for Critical Points

By factoring out the common term \(4x\), we get:

\[4x(x^2 - 3x - 4) = 0\]

Solving for \(x^2 - 3x - 4 = 0\), we find the solutions for \(x\).

Calculating the Solutions

Using the quadratic formula, we find the solutions for \(x\):

\[x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-4)}}{2(1)}\]

This gives us the solutions \(x = -1\) and \(x = 4\).

Testing for Maximum and Minimum

Now, we need to test the intervals around the critical points to determine whether they correspond to maximum or minimum values. We also need to check the function's values at the endpoints of the given interval \([-2, 0]\).

Calculating the Function Values

We calculate the function values at the critical points and the endpoints:

- \(f(-2)\) - \(f(0)\) - \(f(-1)\) - \(f(4)\)

Determining the Maximum and Minimum Values

By comparing the function values at these points, we can determine the maximum and minimum values of the function \(f(x) = x^4 - 4x^3 - 8x^2 + 3\) on the interval \([-2, 0]\).

I hope this helps! Let me know if you need further assistance.