The function f(x) = ln(x) + x has a minimum at:

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Question: The function f(x) = ln(x) + x has a minimum at:

Options:

Correct Answer: x = 1

Solution:

Finding f\'(x) = 1/x + 1. Setting f\'(x) = 0 gives x = 1 as the minimum point.

The function f(x) = ln(x) + x has a minimum at:

The function f(x) = ln(x) + x has a minimum at:

Step 1: Write down the function f(x) = ln(x) + x.
Step 2: Find the derivative of the function, f'(x). The derivative of ln(x) is 1/x and the derivative of x is 1, so f'(x) = 1/x + 1.
Step 3: Set the derivative equal to zero to find critical points: 1/x + 1 = 0.
Step 4: Solve for x. Rearranging gives 1/x = -1, which means x = -1. However, since ln(x) is only defined for x > 0, we ignore this solution.
Step 5: Check the behavior of f'(x) around x = 1. For x < 1, f'(x) is negative, and for x > 1, f'(x) is positive, indicating a minimum at x = 1.
Step 6: Conclude that the function f(x) has a minimum at x = 1.

Differentiation – Understanding how to find the derivative of a function to identify critical points.
Critical Points – Identifying where the derivative equals zero to find potential minima or maxima.
Second Derivative Test – Using the second derivative to confirm whether a critical point is a minimum or maximum.