f(x)=xlnx Find the minimum value
To find the minimum value of the function f(x) = xlnx, we can start by taking the derivative of f(x) with respect to x. The minimum value occurs where the derivative is equal to zero or does not exist.
Step 1: Take the derivative of f(x) = xlnx using the product rule.
f'(x) = (1)(lnx) + (x)(d/dx[lnx])
= lnx + x(1/x)
= lnx + 1
Step 2: Set the derivative equal to zero and solve for x.
lnx + 1 = 0
lnx = -1
Step 3: Exponentiate both sides using base e.
e^(lnx) = e^(-1)
x = 1/e
Step 4: Check for the second derivative to ensure it is a minimum.
Take the derivative of f'(x) with respect to x.
f''(x) = (d/dx[lnx]) + (d/dx[1])
= (1/x) + 0
= 1/x
Since the second derivative, f''(x) = 1/x, is always positive for x > 0, the critical point x = 1/e is indeed the minimum value of the function.
Therefore, the minimum value of f(x) = xlnx is f(1/e) = (1/e)ln(1/e).
To find the minimum value of the function f(x) = xln(x), we need to take the derivative and set it equal to zero.
1. Take the derivative of f(x) with respect to x using the product rule and chain rule:
f'(x) = (1 * ln(x)) + (x * 1/x) = ln(x) + 1
2. Set the derivative equal to zero to find critical points:
ln(x) + 1 = 0
3. Subtract 1 from both sides:
ln(x) = -1
4. Take the exponential of both sides to eliminate the natural logarithm:
e^(ln(x)) = e^(-1)
x = 1/e
Therefore, the critical point/candidate for the minimum value is x = 1/e.
We need to check the value of f(x) at this critical point to confirm it is indeed the minimum.
5. Substitute x = 1/e into the original function f(x):
f(1/e) = (1/e) * ln(1/e)
= (1/e) * (-1)
= -1/e
So, the minimum value of the function f(x) = xln(x) is -1/e.