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A local minimum is a point where a function's value is lower than at any nearby points. It's crucial to understand that a local minimum isn't necessarily the lowest value over the entire range of the function. Rather, it's just lower than any other value close to it.

Imagine hiking up a hill and descending into a small valley before climbing again. That small valley represents a local minimum. Mathematically, if you have a function, say, \(f(x)\), and there's a point \(x = a\) such that \(f(a)\) is less than the values of all close-by points, then \(f(x)\) has a local minimum at \(x = a\). A clear sign of this is the change in the sign of the slope (or the first derivative) of the function from negative to positive as you pass through \(a\).

A critical point of a function is where its first derivative is zero or undefined. These points are essential as they help identify potential local maxima and minima.

Consider the function \(f(x)\). If at \(x = a\), the derivative \(f^{\rprme}(a) = 0\) or \(f^{\rprme}(a)\) doesn’t exist, then \(a\) is a critical point. These points are where the function either changes direction or has a horizontal tangent. Critical points don't automatically mean the function has a local minimum or maximum; they could also be points of inflection where the curve shifts shape.

In our exercise, we know that \(f^{\rprme}(a) = 0\), which confirms that \(a\) is indeed a critical point.

An increasing function is one in which the function values get larger as you move from left to right across the graph. This is formally described by saying that for any two points \(x_1\) and \(x_2\) where \(x_1 < x_2\), the function value at \(x_2\) is greater than the function value at \(x_1\); in other words, \(f(x_1) < f(x_2)\).

If the derivative of a function \(f^{\rprme}(x)\) is positive on an interval, the original function \(f(x)\) is increasing on that interval. In our problem, we know that \(f^{\rprme}(x)\) is increasing at \(x = a\). This implies that near the point \(a\), the slope of \(f(x)\) starts to go from negative (downhill) to positive (uphill), indicative of a local minimum at \(x = a\).

The first derivative of a function \( f(x) \), denoted as \( f^{\rprme}(x) \), measures the rate at which \( f(x) \) changes as \( x \) changes. Simply put, it gives the slope of the tangent line to the function’s graph at any given point. This slope tells you whether the function is increasing or decreasing at that point.

When the first derivative \( f^{\rprme}(x) \) is positive, the function \( f(x) \) is increasing. Conversely, when \( f^{\rprme}(x) \) is negative, the function is decreasing. If \( f^{\rprme}(a) = 0 \), it suggests a potential local minimum or maximum at \( x = a \).

In our context, the given details tell us \( f^{\rprme}(a) = 0 \) and \( f^{\rprme}(x) \) is increasing at \( x = a \), which confirms that \( f(x) \) must have a local minimum at \( x = a \) through the first derivative test.