If f(x) = ln(x) + x^2, then the function is increasing for:

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Question: If f(x) = ln(x) + x^2, then the function is increasing for:

Options:

Correct Answer: x > 0

Solution:

The derivative f\'(x) = 1/x + 2x. For f\'(x) > 0, we need 1/x + 2x > 0, which holds for x > 0.

If f(x) = ln(x) + x^2, then the function is increasing for:

If f(x) = ln(x) + x^2, then the function is increasing for:

Step 1: Identify the function f(x) = ln(x) + x^2.
Step 2: Find the derivative of the function, which is f'(x).
Step 3: The derivative f'(x) is calculated as f'(x) = 1/x + 2x.
Step 4: To determine when the function is increasing, we need to find when f'(x) > 0.
Step 5: Set up the inequality: 1/x + 2x > 0.
Step 6: Analyze the inequality. Since 1/x is positive for x > 0 and 2x is also positive for x > 0, the entire expression is positive for x > 0.
Step 7: Conclude that the function f(x) is increasing for x > 0.

Derivative and Monotonicity – Understanding how the derivative of a function indicates whether the function is increasing or decreasing.
Natural Logarithm Properties – Knowledge of the domain of the natural logarithm function and its behavior.
Quadratic Functions – Recognizing the behavior of quadratic functions and their impact on the overall function's growth.