The maximum value of the function `f(x)=(1+x)^(0.3)/(1+x^(0.3))` in [0,1] is

Options

A. 1
B. `2^(0.7)`
C. `2^(-0.7)`
D. `2^(0.3)`

Answer

To find the maximum value of the function \( f(x) = \frac{(1+x)^{0.3}}{1+x^{0.3}} \) in the interval \([0, 1]\), we will follow these steps:

### Step 1: Find the derivative of \( f(x) \)
To find the critical points, we need to differentiate \( f(x) \) with respect to \( x \).

Using the quotient rule:
\[ f'(x) = \frac{(1+x^{0.3}) \cdot \frac{d}{dx}[(1+x)^{0.3}] - (1+x)^{0.3} \cdot \frac{d}{dx}[1+x^{0.3}]}{(1+x^{0.3})^2} \]

Calculating the derivatives:
- The derivative of \( (1+x)^{0.3} \) is \( 0.3(1+x)^{-0.7} \).
- The derivative of \( 1+x^{0.3} \) is \( 0.3x^{-0.7} \).

Plugging these into the derivative:
\[ f'(x) = \frac{(1+x^{0.3}) \cdot 0.3(1+x)^{-0.7} - (1+x)^{0.3} \cdot 0.3x^{-0.7}}{(1+x^{0.3})^2} \]

### Step 2: Set the derivative equal to zero
To find the critical points, we set \( f'(x) = 0 \):
\[ (1+x^{0.3}) \cdot 0.3(1+x)^{-0.7} - (1+x)^{0.3} \cdot 0.3x^{-0.7} = 0 \]

Dividing through by \( 0.3 \) (since it is non-zero):
\[ (1+x^{0.3}) (1+x)^{-0.7} = (1+x)^{0.3} x^{-0.7} \]

### Step 3: Simplify the equation
Rearranging gives:
\[ (1+x^{0.3}) = (1+x) x^{-0.7} (1+x)^{0.7} \]
This simplifies to:
\[ (1+x^{0.3}) = (1+x) \]

### Step 4: Solve for \( x \)
Setting the equation \( 1 + x^{0.3} = 1 + x \) leads to:
\[ x^{0.3} = x \]

This implies:
\[ x^{0.3 - 1} = 1 \implies x^{-0.7} = 1 \implies x = 1 \]

### Step 5: Evaluate \( f(x) \) at critical points and endpoints
Now we evaluate \( f(x) \) at the endpoints \( x = 0 \) and \( x = 1 \):

1. For \( x = 0 \):
\[ f(0) = \frac{(1+0)^{0.3}}{1+0^{0.3}} = \frac{1}{1} = 1 \]

2. For \( x = 1 \):
\[ f(1) = \frac{(1+1)^{0.3}}{1+1^{0.3}} = \frac{2^{0.3}}{1+1} = \frac{2^{0.3}}{2} = 2^{-0.7} \]

### Step 6: Compare values
Now we compare the values:
- \( f(0) = 1 \)
- \( f(1) = 2^{-0.7} \)

Since \( 2^{-0.7} \) is less than 1, the maximum value in the interval \([0, 1]\) is:
\[ \text{Maximum value} = 1 \]

### Final Answer
The maximum value of the function \( f(x) \) in the interval \([0, 1]\) is \( 1 \).

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