Mention the properties to be covered: zeros, symmetry, derivatives, extrema, and concavity.
1. Zeros: Zeros or roots of a function are the values of the input variable where the output of the function equals zero. These are the points where the graph of the function intersects the x-axis.
2. Symmetry: Symmetry refers to the property of a function that remains unchanged when it is reflected or rotated. There are different types of symmetry like even symmetry (reflection across the y-axis) and odd symmetry (reflection across the origin).
3. Derivatives: The derivative of a function represents its rate of change at any given point. It gives the slope of the tangent line to the graph of the function at a particular point. Derivatives are used to analyze the behavior of a function and can provide information about increasing/decreasing intervals and where the function reaches its maximum or minimum values.
4. Extrema: Extrema refer to the maximum and minimum values of a function. A function may have local extrema (maximum or minimum within a specific interval) and global extrema (maximum or minimum over the entire domain).
5. Concavity: Concavity describes the curvature of a function's graph. A function can be concave up (convex), where it curves upward like a U-shape, or concave down (concave), where it curves downward like an inverted U-shape. The second derivative of a function is used to determine its concavity.