Quadratic Functions

We can write a quadratic function in explicit form as v = ax2 + bx + c. x e R, u £ 0

As figure 2.24 shows, this is a useful function in economics because in its convex form, with a > 0. it could be used to depict a typical U-shaped average or marginal-cost curve, while in its concave form, with a < 0, it could depict a typical total-revenue or total-profit curve. (Note that in these examples the domain of the function must be restricted to R+, since negative outputs arc not allowed.) The unique minimum (<n the convex case) or maximum (in the concave case) always occurs at the point x* = —b/2a. Thus, if we want a function to have a maximum at a positive value of v. we must choose b > 0. while if we want a function to have a minimum at a positive value of x, we must choose /< < 0. Finally, the value of c will determine whether y is positive, negative, or zero at this maximum or minimum.

Figure 2.24 Quadratic functions
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