Question

F(x)=-x^2 на [-1;2]f(x)=3x-x^3 на [-3;0]

Answer 1

Первый вопрос овет 56
вторрй вопрос ответ 23

Answer 2

To find the absolute maximum and minimum values of the given functions on their respective intervals, we need to follow these steps:

1. Find the critical points within the interval.
2. Evaluate the function at the critical points and endpoints of the interval.
3. Compare the values obtained in step 2 to determine the absolute maximum and minimum.

Let's start with the first function:

1. Function: $f(x) = -x^2$ on the interval $[-1, 2]$

To find critical points, we need to find where the derivative of $f(x)$ is equal to zero or undefined.

$f'(x) = -2x$

Critical point: Set $f'(x) = 0$

$-2x = 0 \Rightarrow x = 0$

The critical point is $x = 0$. Since the interval $[-1, 2]$ is closed and bounded, we also need to check the function at the endpoints of the interval.

Evaluate $f(x)$ at the endpoints and critical point:

$f(-1) = -(-1)^2 = -1$

$f(0) = -(0)^2 = 0$

$f(2) = -(2)^2 = -4$

Now we compare the function values:

• Absolute maximum: $f(0) = 0$
• Absolute minimum: $f(2) = -4$

Next, let's move on to the second function:

2. Function: $f(x) = 3x - x^3$ on the interval $[-3, 0]$

To find critical points, we need to find where the derivative of $f(x)$ is equal to zero or undefined.

$f'(x) = 3 - 3x^2$

Critical points: Set $f'(x) = 0$

$3 - 3x^2 = 0 \Rightarrow 3x^2 = 3 \Rightarrow x^2 = 1 \Rightarrow x = \pm 1$

The critical points are $x = -1$ and $x = 1$. Again, since the interval $[-3, 0]$ is closed and bounded, we also need to check the function at the endpoints of the interval.

Evaluate $f(x)$ at the endpoints and critical points:

$f(-3) = 3(-3) - (-3)^3 = 9$

$f(-1) = 3(-1) - (-1)^3 = -4$

$f(0) = 3(0) - (0)^3 = 0$

Now we compare the function values:

• Absolute maximum: $f(-3) = 9$
• Absolute minimum: $f(-1) = -4$

So, for the function $f(x) = -x^2$ on the interval $[-1, 2]$, the absolute maximum is 0, and the absolute minimum is -4.

And for the function $f(x) = 3x - x^3$ on the interval $[-3, 0]$, the absolute maximum is 9, and the absolute minimum is -4.