1. Applications to Equations
1. Number of Real Roots of an Equation
1. The number of distinct real roots of the equation is equal to the number of intersections between the graph of the function and the -axis.
2. The number of distinct real roots of the equation is equal to the number of intersections between the graphs of the two functions and . Alternatively, this can be determined by finding the number of intersections between the graph of and the -axis.


2. Determining the Roots of a Cubic Equation
   When a cubic function has local extrema, the roots of the cubic equation can be determined using the extrema as follows:
1. If (local maximum) (local minimum) , there are three distinct real roots.
2. If (local maximum) (local minimum) , there is one real root and one repeated root (i.e., two distinct real roots).
3. If (local maximum) (local minimum) , there is one real root and two complex roots.
4. If the cubic function does not have local extrema, the equation either has a triple root or one real root and two complex roots.


2. Applications to Inequalities
1. To prove that the inequality holds in a given interval for a function , it is sufficient to show that the minimum value of in that interval is greater than or equal to .