Find all the values of the parameter a(a ≤1) for which the...

Find all the values of the parameter $a(a \leq 1)$ for which the area of the figure bounded by the pair of straight lines $y^{2}-3 y+2=0$ and the curves $y=[a] x^{2}, y=\frac{1}{2}[a] x^{2}$ is
the greatest, where [.] denotes greatest integer function.

Key Concepts

Greatest Integer Function

The greatest integer function, also known as the floor function, assigns to each real number the largest integer less than or equal to it. In the context of the problem, it is used to convert the continuous parameter a into an integer, which in turn affects the coefficients of the quadratic curves, altering their shapes and positions and influencing the area enclosed by these curves.

Area Between Curves

The concept of finding the area between curves involves setting up an integral over a region bounded by curves whose intersection points determine the limits of integration. In this problem, the area is calculated between two parabolic curves, where the integration requires determining the points at which the curves intersect with each other or with the given lines.

Intersection of Curves and Lines

Identifying the intersection points of the curves and the pair of straight lines is crucial to determine the region over which the area is calculated. This process generally involves solving equations simultaneously, allowing one to define precise integration limits for subsequent area computation.

Parameter Optimization

Parameter optimization involves analyzing how a variable parameter—in this case, the value a (subject to a ? 1) that affects the coefficients via the greatest integer function—impacts a desired quantity, such as an area. The goal is to find the parameter value(s) that maximize or minimize this quantity, often by considering different cases or by studying the functional dependence on the parameter.

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