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To understand how a function behaves, we can analyze where it is increasing or decreasing. These are critical aspects of differential calculus. A function is said to be increasing on an interval if, as we move from left to right (along the x-axis), the function values (on the y-axis) rise. Conversely, a function is decreasing if these values fall as we move in the same direction.

Determining these intervals involves the function's derivative. The derivative indicates the slope of the tangent line at any point. If the derivative is positive on an interval, the function is increasing there. If it is negative, then the function is decreasing.

• When the derivative changes from negative to positive, there could be a local minimum.
• When it changes from positive to negative, a local maximum might occur.

Recognizing these changes helps in sketching and understanding the overall behavior of a function.

When dealing with functions involving absolute values, extra care is needed. The absolute value function has a sharp point at zero, which affects its differentiation. To find derivatives for such functions, we must consider the function's behavior on either side of zero (positive and negative values).

For \(f(x) = 2x^2 - \ln |x|\), the presence of \(|x|\) in the logarithmic term requires considering two cases:

• When \(x > 0\), treat \(|x|\) as \(x\), so the derivative of \(-\ln x\) is \(-1/x\).
• When \(x < 0\), treat \(|x|\) as \(-x\), making the derivative \(1/x\) since \(-\ln(-x) = \ln(x)\) by the definition of natural logarithms.

This split ensures correct differentiation across the whole domain, especially at points where \(x = 0\) is undefined, emphasizing why absolute value functions can be tricky.

Interval analysis is a useful tool in calculus for understanding where functions change their behavior. It involves checking the signs of the derivative over different intervals. This process helps determine where a function is increasing or decreasing.

Consider \(f'(x) = 4x - 1/x\) for \(x > 0\) and \(f'(x) = 4x + 1/x\) for \(x < 0\). To find where these expressions are positive or negative, we solve:

• For \(x < 0\), solving \(4x + 1/x > 0\) gives \(x < -1/4\).
• For \(x > 0\), solving \(4x - 1/x > 0\) gives \(x > 1/4\).

After determining where the derivative is positive or negative, we can conclude which intervals display increasing or decreasing behavior. Through this approach, interval analysis becomes a powerful method for studying the intricacies of functions in calculus.