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In calculus, a local minimum of a function is a point where the function value is lower than at any nearby points, making it a 'valley' in the graph of the function. To identify a local minimum, two main conditions must be checked using derivatives.

First, we need the first derivative to equal zero at that point. This condition indicates that the slope of the tangent to the graph at that point is horizontal, which is typical for both local minima and maxima. Mathematically, for a function f(x), if x=c is a point of local minimum, then f'(c)=0.

However, this condition alone is not enough; we also require the second derivative to be greater than zero at that point. A positive second derivative implies that the function is 'curving upwards,' confirming a local minimum. If the second derivative at that point were negative, the point would instead be a local maximum. In our exercise, by setting the first derivative of the function equal to zero at x=2 and ensuring the second derivative is positive, we found that a=2 meets the criteria for a local minimum at that point.

A point of inflection is a particular point on a curve at which the direction of curvature changes. This means the curve switches from 'bending' upwards to downwards, or vice versa. Inflection points are significant as they mark the transition in the concavity of the graph.

To determine if a given point is an inflection point, one commonly used criterion is setting the second derivative of the function to zero at that point. When the second derivative f''(x) is zero at x=c, it suggests a possible point of inflection. Nevertheless, a second derivative equaling zero is a necessary but not a sufficient condition for an inflection point, as not all such points are inflection points. It's often further validated by checking the sign change in the second derivative on either side of the point.

In the given exercise, we found the second derivative and set it equal to zero at x=1. Solving this provided us the value of a=-1, which indicates the parameter value that makes the function f(x) have a point of inflection at x=1.

The first derivative of a function represents the rate at which the function's value is changing at any point, essentially measuring the slope of the tangent line to the function's graph at that point. It's a fundamental tool in calculus for analyzing the behavior of functions, such as identifying increasing/decreasing intervals and finding local extremes.

In practical terms, if the first derivative f'(x) is positive at a point x, it means the function is increasing at that point. Conversely, if f'(x) is negative, the function is decreasing. The point where the first derivative is zero can be a candidate for a local minimum or maximum, pending further investigation with other tests like the second derivative test.

For our problem, by finding the first derivative and setting it to zero at x=2, we evaluated part of the criteria for a local minimum. This step is pivotal for understanding how the function's growth is halted momentarily at this specific point.

The second derivative gives us an insight into the curvature or concavity of the function's graph. It describes how the slope of the tangent line, found by the first derivative, is changing at any given point. This becomes particularly useful for identifying local minima and maxima, as well as inflection points.

More specifically, if the second derivative f''(x) is positive, the graph is concave up, resembling the shape of a bowl. If it’s negative, the graph is concave down, like an upside-down bowl. When the second derivative is zero, it's indicative of a potential inflection point, where the concavity might change.

In the problem presented, we used the second derivative to validate the existence of a local minimum at x=2 (by confirming it's positive) and to find an inflection point at x=1 (by setting it to zero). The second derivative is thus a powerful tool in the analysis of a function's behavior.