Number of Real Roots of an Equation

1. For the equation :
   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. To determine the number of real roots for , first sketch the graph of using its derivative.
   
   
3. For the equation :
1. The number of distinct real roots is the number of intersections between the graphs of the functions and .
2. This is also equivalent to the number of intersections between the graph of and the -axis (i.e., solving ).

Application to Inequalities

1. To prove that the inequality holds for all real values of :
   Show that the minimum value of the function is greater than or equal to .
2. To prove that the inequality holds within a specific interval:
   Show that the minimum value of in that interval is greater than or equal to .
3. To prove that the inequality holds within a specific interval:
   Define a new function , and show that the minimum value of in that interval is greater than or equal to .