Maxima and minima





 

It is obvious from the diagram that the function y=f(x) is maximum at P and is minimum at Q.

 

At both these points tangent is parallel to x-axis, so that its slope is zero.

Maxima and minima

Maxima and minima


\therefore \dfrac{dx}{dy} = 0 \, \, \, \, or \, \, f' (x) = 0 for both maximum and minimum.

 

Working Rule

 

(1) Find out \dfrac{dx}{dy} from the equation f(x,y) = c and put \dfrac{dx}{dy} -0 . Solve it for x. Let, on solving x=a, x=b.

 

(2) If \dfrac{d^2 y}{dx^2} < 0 \, \, \, or \, \, \, \dfrac{d^2y}{dx^2} = negative for x=1, then y is maximum at x=a.

If \dfrac{d^y}{dx^2} > 0 \, \, or \, \, \dfrac{d^2y}{dx^2} = positive for x=b, then y is minimum at x=b.

 

(3) If both \dfrac{dx}{dy} \, \, and \, \, \dfrac{d^2y}{dx^2} are 0 for some value of x, we have to find \dfrac{d^3 y}{dx^3} . If it is 0 then f(x) is maximum if \dfrac{d^4y}{dx^4} <0 , minimum if \dfrac{d^4 y}{dx^4} > 0 .

However, if \dfrac{d^2 y}{dx^2} is zero for some x and \dfrac{d^3 y}{dx^3} = 0 for that x then f(x) is neither minimum nor maximum for that x. It is called the point of inflexion. At such point the curve changes from concave to convex or from convex to concave.

 

(4) Between two minimum there is at least one maximum. Similarly between to maxima, there is at least one minima.

Maxima and minima occur alternatively.

 

(5) At a maxima, \dfrac{dx}{dy} changes sign from positive to negative and at a minima, \dfrac{dy}{dx} changes sign from negative to positive.

 

Concavity and Convexity

 

A curve is convex and concave at some point ‘P’ to x-axis according as: y \dfrac{d^2 y}{dx^2} is positive or negative at P.

 

 

Critical points

 

(1) The values of x for which f’ (x) = 0 are called stationary values or critical values of x and corresponding values of f(x) are called stationary or turning values of f(x).

 

(2) if f(x) is not differentiable at x=a, then also x=a is called critical points. Also f’ (x) = 0 has no solution in an interval (a,b) then (a,b) is called range of those points where critical points do not exist.

 

Important point:

 

(1) Maximum and minimum values of a \, cos \, \theta \pm sin \theta \, \, is \, \, \sqrt{ (a^2 + b^2 )} and \, \, - \sqrt{ ( a^2 + b^2 ) } .

 

(2) Least value of x+y subject to condition xy=constant, occurs when x=y.



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