For the function: $$ f(x)=4 x-6 x^{2 / 3} $$ on the interval $$ [-1,3] $$, The absolute min occurs at $$ (-1,-10) $$ The absolute max occurs at $$ ([?],[~]) $$

Question

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To find the absolute maximum of the function $$ f(x) = 4x - 6x^{2/3} $$ on the interval $$ [-1, 3] $$, we need to follow these steps:

1. **Evaluate the function at the endpoints of the interval.**
2. **Find the critical points by taking the derivative and setting it to zero.**
3. **Evaluate the function at the critical points within the interval.**
4. **Compare the values obtained from the endpoints and critical points to determine the absolute maximum.**

### Step 1: Evaluate at the Endpoints

First, we evaluate $$ f(x) $$ at the endpoints $$ x = -1 $$ and $$ x = 3 $$.

- For $$ x = -1 $$:
  $$
  f(-1) = 4(-1) - 6(-1)^{2/3} = -4 - 6(1) = -4 - 6 = -10
  $$

- For $$ x = 3 $$:
  $$
  f(3) = 4(3) - 6(3)^{2/3} = 12 - 6(3^{2/3})
  $$
  To calculate $$ 3^{2/3} $$:
  $$
  3^{2/3} = (3^{1/3})^2 = (\sqrt[3]{3})^2
  $$
  We can approximate $$ 3^{1/3} \approx 1.442 $$, so:
  $$
  3^{2/3} \approx (1.442)^2 \approx 2.08
  $$
  Thus,
  $$
  f(3) \approx 12 - 6(2.08) = 12 - 12.48 \approx -0.48
  $$

### Step 2: Find Critical Points

Next, we find the derivative of $$ f(x) $$:
$$
f'(x) = 4 - 6 \cdot \frac{2}{3} x^{-1/3} = 4 - 4x^{-1/3}
$$
Setting the derivative to zero to find critical points:
$$
4 - 4x^{-1/3} = 0 \implies 4x^{-1/3} = 4 \implies x^{-1/3} = 1 \implies x = 1
$$

### Step 3: Evaluate at the Critical Point

Now we evaluate $$ f(x) $$ at the critical point $$ x = 1 $$:
$$
f(1) = 4(1) - 6(1)^{2/3} = 4 - 6(1) = 4 - 6 = -2
$$

### Step 4: Compare Values

Now we compare the values:
- $$ f(-1) = -10 $$
- $$ f(3) \approx -0.48 $$
- $$ f(1) = -2 $$

The maximum value occurs at $$ x = 3 $$ with $$ f(3) \approx -0.48 $$.

### Conclusion

The absolute maximum occurs at $$ (3, -0.48) $$.

Thus, the final answer is:
The absolute max occurs at $$ (3, -0.48) $$.
The absolute maximum occurs at $$ (3, -0.48) $$.

For the function:

on the interval ,
The absolute min occurs at
The absolute max occurs at

Upstudy AI Solution

Answer

Solution

Step 1: Evaluate at the Endpoints

Step 2: Find Critical Points

Step 3: Evaluate at the Critical Point

Step 4: Compare Values

Conclusion

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