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Critical numbers play a crucial role in understanding a function's behavior. They are the points where the derivative of a function is either zero or undefined. These numbers help pinpoint potential turning points or points where the function changes direction.

For the function given, \( f(x) = x^4 - 32x + 4 \), its first derivative is \( f'(x) = 4x^3 - 32 \). To find critical numbers, we set \( f'(x) = 0 \). Solving \( 4x^3 - 32 = 0 \) gives \( x^3 = 8 \), leading to \( x = 2 \).

This tells us that \( x = 2 \) is a critical number and suggests that around this point, the function might change from increasing to decreasing or vice versa.

Identifying where a function is increasing is essential for understanding its overall behavior. A function is increasing when its derivative is positive. This positivity implies the slope is upward as you move along the curve.

For the function \( f(x) = x^4 - 32x + 4 \), we've established that its derivative is \( f'(x) = 4x^3 - 32 \). To determine the increasing intervals, look at where \( f'(x) > 0 \).

Solving this inequality for \( f'(x) \), we find that for \( x > 2 \), \( f'(x) \) is positive. Hence, the function is increasing on the interval \( (2, \infty) \). This means as \( x \) grows beyond 2, the value of \( f(x) \) keeps rising.

Understanding decreasing intervals of a function gives insight into where the function values are dropping. A function decreases in intervals where its derivative is negative, meaning the slope of the tangent line to the graph is downward.

In our example, with \( f(x) = x^4 - 32x + 4 \), the derivative \( f'(x) = 4x^3 - 32 \) helps determine these intervals. By analyzing \( f'(x) < 0 \), we can identify where the function descends.

When evaluating \( f'(x) \), we discover that for \( x < 2 \), \( f'(x) \) is negative. Therefore, the function is decreasing on the interval \( (-\infty, 2) \). This indicates that as \( x \) approaches 2 from the left, \( f(x) \) decreases.

A relative extremum is a point where the function changes from increasing to decreasing or vice versa, resulting in a relative maximum or minimum. The First Derivative Test provides a systematic approach to find these points.

By using the First Derivative Test on our function \( f(x) = x^4 - 32x + 4 \), we examine the change in the sign of the derivative \( f'(x) = 4x^3 - 32 \) around the critical number \( x = 2 \).

Before \( x = 2 \), \( f'(x) \) is negative, and after \( x = 2 \), \( f'(x) \) becomes positive. This sign change indicates a transition from decreasing to increasing, showing a relative minimum at \( x = 2 \). Hence, the function reaches a low point at this critical number, which reveals a critical characteristic of its curve.