Page 151 - DECO403_MATHEMATICS_FOR_ECONOMISTS_ENGLISH

Mathematics for Economists

Note dy
2
3. For …………………., at x = a, the value of = negative value.
dx 2
2
dy
4. For minimum at x = a, the value of = is …………….. value.
dx 2
9.4 Conditions to Absence of Maxima or Minima

At point x = a, the value of function y = f(x) will neither be maximum nor minimum if
2
3
dy dy
The value of = 0 and value of ≠≠ ≠≠ ≠ 0
dx 2 dx 3
Properties of Maximum and Minimum value

1. Maximum value comes after minimum value and minimum comes after maximum viz
maximum and minimum comes in a sequence.
2. There will be a certain maximum or minimum value between the two equal values of
function.
3. At point touching lines are parallel to x-axis where the maximum and minimum of function
dy
are there. Therefore, at such points value of will be 0, solving the equation after putting
dx
dy
dx = 0, value of x can be obtained, over which the value of the function is maximum or
minimum.
dy
4. At the maximum or minimum point of function the sign of changes. At maximum point
dx
it becomes negative from positive and contrary to this it becomes positive from negative.

Task Define the minimum.

9.5 Steps for Finding Maxima and Minima of the Function y = f(x)

dy
(i) Calculating of y = f(x)
dx
dy
(ii) Determining various values of x from the equation obtain by assigning =0
dx
(iii) Assume the different values of x are a , a , a etc.
1 2 3
2
dy
2
2
(iv) Obtaining, d y/dx finding the value of on a , a , a etc.
dx 2 1 2 3
2
dy
If for any value of x value of is positive, then function for that value of x is minimum and
dx 2
if it is negative, then value would be maximum.

144 LOVELY PROFESSIONAL UNIVERSITY