Question

Найти наибольшее и наименьшее значение функции f(x)=x^3-6x^2+9x-4 на отрезке [0;2]

Answer 1

Ответ на картинке внизу страницы

Answer 2

To find the maximum and minimum values of the function $f(x) = x^3 - 6x^2 + 9x - 4$ on the interval $[0, 2]$, we need to check the critical points and the endpoints of the interval.

Step 1: Find the critical points: Critical points are the points where the derivative of the function is equal to zero or undefined. We'll start by finding the derivative of $f(x)$ and then set it to zero to find the critical points.

$f(x) = x^3 - 6x^2 + 9x - 4$

$f'(x) = 3x^2 - 12x + 9$

Now, let's set $f'(x) = 0$ to find the critical points:

$3x^2 - 12x + 9 = 0$

Dividing the equation by 3 to simplify:

$x^2 - 4x + 3 = 0$

Now, let's solve this quadratic equation using factoring:

$(x - 1)(x - 3) = 0$

This equation has two solutions: $x = 1$ and $x = 3$.

Step 2: Check the endpoints of the interval: The interval is $[0, 2]$, so we need to evaluate the function at the endpoints:

$f(0) = 0^3 - 6(0)^2 + 9(0) - 4 = -4$

$f(2) = 2^3 - 6(2)^2 + 9(2) - 4 = 2$

Step 3: Evaluate the function at the critical points: We found two critical points: $x = 1$ and $x = 3$.

$f(1) = 1^3 - 6(1)^2 + 9(1) - 4 = 0$

$f(3) = 3^3 - 6(3)^2 + 9(3) - 4 = 2$

Step 4: Compare the function values: Now, we need to compare the function values at the endpoints and critical points to find the maximum and minimum values on the interval.

$f(0) = -4$

$f(1) = 0$

$f(2) = 2$

$f(3) = 2$

So, the minimum value of the function on the interval $[0, 2]$ is -4, and the maximum value is 2.