(a) Let f(x) = x^(1/x) for all x>/=1. Find the greatest value of f(x).
(b) By (a), find a possitive interger m, such that m^(1/m) >/= n^(1/n) for all positive intergers n.
To find the greatest value of f(x) for f(x) = x^(1/x), we need to find the critical points by taking the derivative of f(x) and setting it equal to zero.
(a) Finding the greatest value of f(x):
Step 1: Find the derivative of f(x).
f(x) = x^(1/x)
ln(f(x)) = ln(x^(1/x))
ln(f(x)) = (1/x) * ln(x)
Differentiate both sides:
(1/f(x)) * f'(x) = (1/x) * ln(x) + (1/x^2)
Simplifying:
f'(x) = f(x) * [(ln(x) + 1)/x^2]
Step 2: Set f'(x) = 0 and solve for x.
f'(x) = 0
f(x) * [(ln(x) + 1)/x^2] = 0
Since f(x) cannot equal zero, we focus on the factor (ln(x) + 1)/x^2:
(ln(x) + 1)/x^2 = 0
ln(x) + 1 = 0
ln(x) = -1
x = e^(-1) = 1/e
Step 3: Determine the maximum value of f(x).
To find the critical points, we need to evaluate f(x) at x = 1/e and at the boundaries (x = 1).
f(1/e) = (1/e)^(1/(1/e)) = (1/e)^e = 1/e^e (approximately 0.6922)
f(1) = 1^1 = 1
Comparing the two values, we can see that f(1/e) = 1/e^e is the maximum value of f(x) when x is greater than or equal to 1.
Therefore, the greatest value of f(x) is approximately 0.6922.
(b) To find a positive integer m that satisfies m^(1/m) ≥ n^(1/n) for all positive integers n, we can use the result from part (a) where the maximum value of f(x) occurs at x = 1/e.
Since we want m^(1/m) ≥ n^(1/n) for all positive integers n, we can take the value of m as the base and 1/m as the exponent.
Thus, m^(1/m) = 1/e^e (the maximum value obtained from part (a)).
Therefore, any positive integer m such that m ≥ 1/e^e will satisfy the inequality m^(1/m) ≥ n^(1/n) for all positive integers n.