Introduction
The application of derivatives plays a very important role in the field of Science engineering, economics, commerce and so on. The problems of the type “the increase in the length of an iron rod due to the increase in temperature”; “the increasing and decreasing tendency of the cost functions”; “the maximum profit to be made with the help of minim investment” and so on can betterly be solved with the help of derivatives.
Local Maxima
A function y = f(x) is said to have the local maximum value or local maxima at x = a , if f(a) > f(a ± h) , for sufficiently small positive value ‘h’.
The local maxima is known as relative maxima of the function.
Local Minima
A function y = f(x) is said to have local minimum value at x= a if f(a) < f(a ± h), for sufficiently small positive value ‘h’.
Absolute Maxima
A function y = f(x) is said to have absolute maximum value or absolute maxima at x = a , if f(a) is greatest of all its valves for all ‘x’ belonging to the domain of the function.
In other word, f(a) is absolute maximum value of f(x) if f(a) ≥ f(x) for all x € D(f).
Procedure to find the Absolute Maxima and Minima
Let y = f (x) be the given function defined in (a, b). Now,
- Find dy/dx or f'(x) i.e first derivative.
- Take f'(x) = 0, to solve for x to get stationary point. Suppose valve of x are ‘c’ and ‘d’.
- Find the value of f(x) at x = a, b, c, d. The least value gives the absolute minimum value and greatest valve gives the absolute maximum value.
Procedure to find Local Maxima and Local Minima
- Let y = f(x) be a function. Then,
- Find dy/dx or f'(x) and f”(x)
- Making f'(x) = 0, to solve for x to get stationary point. Let x = a be one of stationary point.
- Find f” (a).
If f”(a) < 0 then f(x) has maximum value at x=a, Maximum valve is f(a).
If f”(a) > 0 then f(x) has minimum value at x=a, Minimum value is f(a).
If f” (a) = 0 then f”‘(a) ≠ 0 then f(x) has neither maximum nor minimum value.