let f be the function defined by f(x)= lnx/x. what is the absolute minimum of f
To find the absolute minimum of the function f(x) = ln(x)/x, we first need to find the critical points of the function.
To do this, we differentiate the function with respect to x:
f'(x) = (1 - ln(x))/x^2.
Next, we set the derivative equal to 0 and solve for x:
(1 - ln(x))/x^2 = 0
1 - ln(x) = 0
ln(x) = 1
x = e.
So the critical point of the function is x = e.
To determine whether this critical point is a minimum, we can use the second derivative test.
Taking the second derivative of f(x), we get:
f''(x) = 2ln(x)^2 - 3/x^3.
Plugging in the critical point x = e, we find:
f''(e) = 1/e.
Since the second derivative at x = e is positive, this means that the function has a local minimum at x = e.
Therefore, the absolute minimum of the function f(x) = ln(x)/x occurs at x = e and is equal to f(e) = 1/e.