The fraction exceeds its `p^(th) ` power by the greatest number possible, where `p geq2` is

Options

A. `((1)/(p))^(1//(p-1))`
B. `((1)/(p))^(p-1)`
C. `p^(1//p-1)`
D. none of these

Explanation

To solve the problem where the fraction exceeds its \( p^{th} \) power by the greatest number possible, we can follow these steps:

### Step 1: Define the function
Let \( y = x - x^p \). This function represents the difference between the fraction \( x \) and its \( p^{th} \) power.

### Step 2: Find the first derivative
To find the maximum value of \( y \), we need to take the derivative of \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = 1 - p x^{p-1} \]

### Step 3: Set the first derivative to zero
For maxima or minima, we set the first derivative equal to zero:
\[ 1 - p x^{p-1} = 0 \]
This implies:
\[ p x^{p-1} = 1 \]

### Step 4: Solve for \( x \)
Rearranging gives:
\[ x^{p-1} = \frac{1}{p} \]
Taking the \( (p-1)^{th} \) root of both sides, we find:
\[ x = \left(\frac{1}{p}\right)^{\frac{1}{p-1}} = \frac{1}{p^{\frac{1}{p-1}}} \]

### Step 5: Find the second derivative
Next, we find the second derivative to confirm that we have a maximum:
\[ \frac{d^2y}{dx^2} = -p(p-1)x^{p-2} \]

### Step 6: Evaluate the second derivative at the critical point
Substituting \( x = \frac{1}{p^{\frac{1}{p-1}}} \) into the second derivative:
\[ \frac{d^2y}{dx^2} = -p(p-1)\left(\frac{1}{p^{\frac{1}{p-1}}}\right)^{p-2} \]
This expression will be negative since \( p \geq 2 \), confirming that we have a maximum.

### Step 7: Calculate the maximum value of \( y \)
Substituting \( x = \frac{1}{p^{\frac{1}{p-1}}} \) back into the original function \( y \):
\[ y = \frac{1}{p^{\frac{1}{p-1}}} - \left(\frac{1}{p^{\frac{1}{p-1}}}\right)^p \]
Simplifying this gives:
\[ y = \frac{1}{p^{\frac{1}{p-1}}} - \frac{1}{p^{\frac{p}{p-1}}} = \frac{1}{p^{\frac{1}{p-1}}} - \frac{1}{p^{\frac{p-1}{p-1}}} \]
\[ y = \frac{1}{p^{\frac{1}{p-1}}} - \frac{1}{p} \]

### Final Result
Thus, the maximum value of \( y \) occurs at \( x = \frac{1}{p^{\frac{1}{p-1}}} \) and can be calculated as shown above.

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