Minimum Value of (p 1/p)

To find the minimum value of the expression (p 1/p), we need to understand the properties of the function and optimize it accordingly.

Properties of the Function

Let's start by analyzing the function.

The expression (p 1/p) can be written as (p^2 + 1)/p.

Now, let's consider the domain of the function.

Since p 0, we can assume that p > 0.

Also, since we are dealing with real numbers, we know that p^2 + 1 > 0 for all values of p.

Therefore, the expression (p^2 + 1)/p is defined for all p > 0.

Optimizing the Function

To find the minimum value of the expression (p 1/p), we can differentiate it with respect to p and set the derivative equal to zero.

(p^2 + 1)/p = p - 1/p^2

Simplifying this equation, we get:

p^3 - p + 1 = 0

This is a cubic equation that can be solved using numerical methods.

However, we can also use some approximation techniques to estimate the root.

Using the intermediate value theorem, we can see that the equation has at least one real root between 0 and 1.

Also, since the coefficient of p^3 is positive and the constant term is positive, we know that the root is greater than 0 and less than 1.

Therefore, we can use the bisection method to approximate the root.

Using this method, we can find that the root is approximately 0.68233.

Therefore, the minimum value of the expression (p 1/p) occurs when p is approximately 0.68233.

At this value of p, the expression evaluates to approximately 1.8205.

Conclusion

The minimum value of the expression (p 1/p) occurs when p is approximately 0.68233, and the expression evaluates to approximately 1.8205.

Therefore, the correct answer is option 'C'.