5.7 Least and greatest function values  (Page 2/3)

Quadratic functions

The graph of quadratic function is known. If coefficient of second power term in the quadratic expression is positive, then function has no maximum and hence no greatest value, whereas its least value is given by “–D/4a”, where D is discriminant of corresponding quadratic equation and “a” is the coefficient of second power term. For x becoming large numbers, function value becomes very large positive value. Therefore, the range of function is given as :

$$\text{Range}=[-\frac{D}{\mathrm{4a}},\infty )$$

On the other hand, if coefficient of highest power term in the quadratic expression is negative, then greatest value of function is given by “–D/4a”, whereas there is no minimum and hence no least value. For x becoming large numbers, function value becomes very large negative value. Therefore, the range of function is given as :

$$\text{Range}=(-\infty ,-\frac{D}{\mathrm{4a}}]$$

We have already discussed these aspects in the module earlier and as such will not elaborate here again.

Monotonic function of same nature

Some functions have same monotonic nature through out domain. Linear polynomial, for example, is either strictly increasing or decreasing. There is no minimum or maximum value. Recall that minimum and maximum values occur between a pair of increasing and decreasing function values. Similar is the case with exponential and logarithmic functions. The ranges of exponential and logarithmic functions are open intervals. Since these functions are strictly increasing or decreasing though out their domain intervals, there is no minimum or maximum. As such, there is no least or greatest values of function.

However, if we investigate these functions in a finite interval, then function values at the boundary of closed interval are the least and greatest values in that interval. Further, functions, which are not singularly monotonic, can also be singly monotonic in a suitably selected interval. For example, sine function is a strictly increasing function in the interval [-π/2, π/2]. Thus, there are two possibilities :

1: Function is monotonic in one type without minimum or maximum function values. In any finite closed interval, however, the function has least and greatest values, which correspond to function values at interval ends.

2: A function is not monotonic of one type, but is rendered monotonic in suitably chosen finite closed interval. The function has least and greatest values, which correspond to function values at interval ends.

We shall use these properties to determine range of a function. In order to understand application of these concepts, we need to read each step of the examples as given here carefully :