\begin{tabular}{l} (1 point) Consider the function $$ f(x)=4-2 x^{2},-4 \leq x \leq 1 $$ \ The absolute maximum value is \ and this occurs at $$ x $$ equals \ The absolute minimum value is \ and this occurs at $$ x $$ equals \ \hline\end{tabular}

Question

UpStudy AI · Accepted Answer

To find the absolute maximum and minimum values of the function $$ f(x) = 4 - 2x^2 $$ on the interval $$ -4 \leq x \leq 1 $$, we will follow these steps:

1. **Find the critical points** by taking the derivative of the function and setting it to zero.
2. **Evaluate the function** at the critical points and the endpoints of the interval.
3. **Determine the absolute maximum and minimum values** from these evaluations.

### Step 1: Find the critical points

First, we calculate the derivative of $$ f(x) $$:

$$
f'(x) = \frac{d}{dx}(4 - 2x^2) = -4x
$$

Next, we set the derivative equal to zero to find the critical points:

$$
-4x = 0 \implies x = 0
$$

### Step 2: Evaluate the function at critical points and endpoints

Now we will evaluate $$ f(x) $$ at the critical point $$ x = 0 $$ and at the endpoints $$ x = -4 $$ and $$ x = 1 $$.

1. **At $$ x = -4 $$**:
   $$
   f(-4) = 4 - 2(-4)^2 = 4 - 2(16) = 4 - 32 = -28
   $$

2. **At $$ x = 0 $$**:
   $$
   f(0) = 4 - 2(0)^2 = 4 - 0 = 4
   $$

3. **At $$ x = 1 $$**:
   $$
   f(1) = 4 - 2(1)^2 = 4 - 2 = 2
   $$

### Step 3: Determine the absolute maximum and minimum values

Now we compare the values obtained:

- $$ f(-4) = -28 $$
- $$ f(0) = 4 $$
- $$ f(1) = 2 $$

From these evaluations, we can conclude:

- The **absolute maximum value** is $$ 4 $$ and it occurs at $$ x = 0 $$.
- The **absolute minimum value** is $$ -28 $$ and it occurs at $$ x = -4 $$.

### Final Answer

The absolute maximum value is $$ 4 $$ and this occurs at $$ x $$ equals $$ 0 $$. The absolute minimum value is $$ -28 $$ and this occurs at $$ x $$ equals $$ -4 $$.
The absolute maximum value is 4, occurring at $$ x = 0 $$, and the absolute minimum value is -28, occurring at $$ x = -4 $$.

(1 point) Consider the function

The absolute maximum value is

and this occurs at equals

The absolute minimum value is

and this occurs at equals

Upstudy AI Solution

Answer

Solution

Step 1: Find the critical points

Step 2: Evaluate the function at critical points and endpoints

Step 3: Determine the absolute maximum and minimum values

Final Answer

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(1 point) Consider the function The absolute maximum value is and this occurs at equals The absolute minimum value is and this occurs at equals