This cookie is set by GDPR Cookie Consent plugin. The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional". The cookie is used to store the user consent for the cookies in the category "Analytics". These cookies ensure basic functionalities and security features of the website, anonymously. Necessary cookies are absolutely essential for the website to function properly. At point L, marginal profit I and thereafter it becomes positive and therefore it will causes the total profit to increase. It will be seen from this figure that up to point L, marginal profit (dπ / dQ) that is, slope of the total profit curve, is negative and has been causing the total profits to fall (in fact, up to L, due to negative marginal profits loses have been increasing. It can be easily known from having a look at Figure 5.10. On the other hand, if the second derivative at a point on a profit function where first derivative is zero is positive, it shows profits are in fact minimum rather than maximum. For example, if the second derivative in our profit function curve is negative, it implies that profits are maximum at the level where first derivative is equal to zero. Since the second derivative of a function when measured at the maximisation level is always negative and when measured at the minimisation level is always positive, it can be used to distinguish between points of maximum and minimum. That is, marginal profit, its second derivative, d 2π / dQ 2 measures slope of the marginal profit function curve. Thus, in case of profit function, whereas first derivative, dπ / dQ measures the slope of the profit function curve. Whereas the first derivative measures the slope of a function, the second derivative measures the slope of the first derivative. It is with the help of the second derivative of a function that we can distinguish between maximum and minimum along a function. However, at point L profits are minimum and at point H profits are maximum. Important thing to note is that at both minimum point L and maximum point H, first order condition, that is, first derivative dπ/dQ be zero is satisfied at both points, L and H, corresponding to OQ 1 and OQ 2 levels of output. It will be seen from Figure 5.10 that point L represents the minimum point and H represents the maximum point of the profit curve. A profit function curve such as the one drawn in Figure 5.10 may have both minimum point and maximum points.
According to the second order condition, for profit maximisation, the second derivative of the profit function must be negative, that is, d 2π / dQ 2 0.Ĭonsider again the case of profit maximisation explained above. output in the present case), we require to apply the second order condition.
To ensure that the derivative is zero at the profit maximising level of the decision variable (i.e. For example, in a profit function, first derivative is equal to zero, both it at maximum and minimum profit levels. For the optimum value, the first derivative being equal to zero is a necessary condition for maximum or minimum, but it is not a sufficient condition. For setting the first derivative of a function equal to zero and solving the resulting equation for the optimum value of the independent variable does not guarantee that optimum value (maximum or minimum as the case may be) will in fact he obtained.
Second Derivative and Second Order Condition for Optimisation :Ī problem arises when we use the first derivative of a function to determine its maximum or minimum value. For this we simply find the first derivative of the profit function and set it equal to zero.Ģ. However, it is easier to use differential calculus to find the profit-maximising output. Graphical analysis cannot tell us easily exactly at what level of output, profits will be maximum, for it takes time to draw a graph and conclude from it.
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