OP Malhotra Maxima and Minima S.Chand ISC Class-12 Maths Solutions Ch-12

OP Malhotra Maxima and Minima S.Chand ISC Class-12 Maths Solutions Ch-12. Step by step Solutions of OP Malhotra SK Gupta, Anubhuti Gangal S.Chand ISC Class-12 Mathematics with Exe-12(a), Exe-12(b), Exe-12(c), Revision Exercise and Chapter Test Questions. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

OP Malhotra Maxima and Minima S.Chand ISC Class-12 Maths Solutions Ch-12

Class:	12th
Subject:	Mathematics
Chapter  :	Ch-12 Maxima and Minima  of Section -A

Board	ISC
Writer	OP Malhotra, SK Gupta, Anubhuti Gangal
Publications	S.Chand Publications 2020-21

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-: Included Topics :- 

Exe-12(a)

Exe-12(b)

Exe-12(c)

Revision Exercise

Chapter Test

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OP Malhotra Maxima and Minima S.Chand ISC Class-12 Maths Solutions Ch-12

Maxima and Minima  :-

Maxima and Minima are one of the most common concepts in differential calculus. A branch of Mathematics called “Calculus of Variations” deals with the maxima and the minima of the functional.

The first variation is defined as the linear part of the change in the functional, and the second part of the variation is defined in the quadratic part. Functional is expressed as the definite integrals which involve the functions and their derivatives.

Local Maxima and Minima :-

We may not be able to tell whether f(b) is the maximum value of f, but we can give some credit to point . We can do this by declaring B as the local maximum for function f . These are also called relative maxima and minima. These local maxima and minima are defined as:

• If f(a) ≤ f(x) for all x in P′s neighborhood (within the distance nearby P , where x=a ), f is said to have a local minimum at x=a.
• If f(a) ≥ f(x) for all in P′s neighborhood (within the distance nearby P, where x=a), f is said to have a local maximum at x=a.

In the above example, B and D are local maxima and A and C are local minima. Local maxima and minima are together referred to as Local extreme.

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Exe-12(a)

OP Malhotra Maxima and Minima S.Chand ISC Class-12 Maths Solutions Ch-12

Question 1:

Find the turning value of the following functions, distinguishing in each case whether the value is a maximum, minimum, or inflexional :

(i)……………..

………………..

Question 2:

If V = 2………….. greatest value of V.

Question 3:

…………………….

…………………….

…………………….

Question 8:

Find the turning value of the function – x³ + 12x² – 5, distinguishing whether the value is a maximum, minimum, or inflexional.

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Exe-12(b)

 Maxima and Minima S.Chand ISC Class-12 Maths Solutions Ch-12

Find the local maximum and local minimum, of any, of the following functions.

Question 1:

x³ – 9x² + 24x – 12

Question 2:

-x³ + 12x² – 5

Question 3:

……………………

……………………

……………………

Question 20:

Find the maximum slope of the curve y = -x……………….

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Exe-12(c)

OP Malhotra Maxima and Minima S.Chand ISC Class-12 Maths Solutions Ch-12

Question 1:

(i) Find two positive number whose ……… as possible.

(ii) Find the …………….. = 8

(iii) Find two …………….. xy is maximum.

(iv) Let x and y be two …………….. of x + y.

Question 2:

Determine two positive real number whose sum is 15 and the sum whose squares is minimum.

Question 3:

…………………..

…………………..

…………………..

Question 27:

Assuming that the stiffness of a beam of rectangular ………………………. lof of diameter a.

Question 28:

Show that the radius of the right circular …………………. half that of the cone.

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Critical Point :-

In mathematics, a Critical pointof a differential function of a real or complex variable is any value in its domain where its derivative is 0. We can hence infer from here that every local extremum is a critical point but every critical point need not be a local extremum. So, if we have a function which is continuous, it must have maxima and minima or local extrema. This means that every such function will have critical points. In case the given function is monotonic, the maximum and minimum values lie at the endpoints of the domain of the definition of that particular function.

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Revision Exercise

OP Malhotra Maxima and Minima S.Chand ISC Class-12 Maths Solutions Ch-12

Question 1:

The sum of three positive numbers is 26. The second number is …………………….. find the numbers.

Question 2:

ABC is a right angled triangle of given area …………….. circle is least.

Question 3:

………………………

………………………

……………………..

Question 16:

A rectangle is inscribed in a semicircle of radius r with one of its sides on the …………………. find the maximum area.

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Chapter Test

OP Malhotra Maxima and Minima S.Chand ISC Class-12 Maths Solutions Ch-12

Question 1:

Find the absolute maximum value and the absolute minimum value for the function

……………………

Question 2:

Find the local maximum ……………………… + 123

Question 3:

Prove that the function ……………….. or minimum.

Question 4:

……………………..

………………………

………………………

……………………….

Question 16:

The function f (x) ………………….. local minimum at

(i) x = 2

(ii) x = -2

(iii) x = 0

(iv) x = 1

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-: End of Maxima and Minima S. Chand ISC Class-12 Maths Solution :-

Return to :-  OP Malhotra S. Chand ISC Class-12 Maths Solutions

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