Let `f(x) = 6x^(4//3) - 3x^(1//3)` defined on `[-1,1]` then

Options

A. maximum value of f is 7
B. maximum value of f is 5
C. maximum value of f is 9
D. minimum value of f is `-3//2`

Answer

To find the maximum and minimum values of the function \( f(x) = 6x^{4/3} - 3x^{1/3} \) defined on the interval \([-1, 1]\), we will follow these steps:

### Step 1: Find the derivative of the function
We start by differentiating \( f(x) \):

\[ f'(x) = \frac{d}{dx}(6x^{4/3}) - \frac{d}{dx}(3x^{1/3}) \]

Using the power rule, we get:

\[ f'(x) = 6 \cdot \frac{4}{3} x^{(4/3) - 1} - 3 \cdot \frac{1}{3} x^{(1/3) - 1} \]

Simplifying this gives:

\[ f'(x) = 8x^{1/3} - x^{-2/3} \]

### Step 2: Set the derivative to zero to find critical points
We set \( f'(x) = 0 \):

\[ 8x^{1/3} - x^{-2/3} = 0 \]

Rearranging this, we have:

\[ 8x^{1/3} = \frac{1}{x^{2/3}} \]

Multiplying both sides by \( x^{2/3} \) (assuming \( x \neq 0 \)):

\[ 8x = 1 \implies x = \frac{1}{8} \]

### Step 3: Evaluate the function at critical points and endpoints
We need to evaluate \( f(x) \) at the critical point \( x = \frac{1}{8} \) and the endpoints \( x = -1 \) and \( x = 1 \).

1. **At \( x = -1 \)**:
\[ f(-1) = 6(-1)^{4/3} - 3(-1)^{1/3} = 6(1) - 3(-1) = 6 + 3 = 9 \]

2. **At \( x = 1 \)**:
\[ f(1) = 6(1)^{4/3} - 3(1)^{1/3} = 6(1) - 3(1) = 6 - 3 = 3 \]

3. **At \( x = \frac{1}{8} \)**:
\[ f\left(\frac{1}{8}\right) = 6\left(\frac{1}{8}\right)^{4/3} - 3\left(\frac{1}{8}\right)^{1/3} \]
First, calculate \( \left(\frac{1}{8}\right)^{1/3} = \frac{1}{2} \) and \( \left(\frac{1}{8}\right)^{4/3} = \left(\frac{1}{2}\right)^{4} = \frac{1}{16} \):
\[ f\left(\frac{1}{8}\right) = 6 \cdot \frac{1}{16} - 3 \cdot \frac{1}{2} = \frac{6}{16} - \frac{3}{2} = \frac{3}{8} - \frac{12}{8} = -\frac{9}{8} \]

### Step 4: Determine the maximum and minimum values
Now we compare the values:
- \( f(-1) = 9 \)
- \( f(1) = 3 \)
- \( f\left(\frac{1}{8}\right) = -\frac{9}{8} \)

The maximum value is \( 9 \) at \( x = -1 \) and the minimum value is \( -\frac{9}{8} \) at \( x = \frac{1}{8} \).

### Final Answer
- Maximum value of \( f(x) \) is \( 9 \).
- Minimum value of \( f(x) \) is \( -\frac{9}{8} \).