f(x)=xlnx Find the minimum value
Take the derivative and determine where it is zero. Then test whether that point is a maximm or a minimum.
The derivative is (ln x + 1)
To find the minimum value of the function f(x) = xlnx, we can use calculus. The minimum value occurs where the derivative of the function is equal to zero.
To find the derivative of f(x), we'll use the product rule:
f'(x) = (1)(lnx) + (x)(1/x)
= lnx + 1
Next, we'll set f'(x) equal to zero and solve for x:
lnx + 1 = 0
Subtracting 1 from both sides:
lnx = -1
Taking the exponential of both sides:
e^(lnx) = e^(-1)
Simplifying:
x = 1/e
So, the critical point occurs at x = 1/e. To determine if it is a minimum or maximum, we can use the second derivative test.
To find the second derivative, we differentiate f'(x):
f''(x) = 1/x
Now, substitute the critical point x = 1/e into f''(x):
f''(1/e) = 1/(1/e)
= e
Since the second derivative is positive (e > 0) at the critical point, x = 1/e, we can conclude that it is a minimum.
Therefore, the minimum value of f(x) = xlnx occurs at x = 1/e.