Question

Find the absolute maximum and absolute minimum values of the function f given by
$straight f left parenthesis straight x right parenthesis space equals sin squared straight x minus cosx comma space straight x space element of space left parenthesis 0 comma space straight pi right parenthesis$

Solution

straight f left parenthesis straight x right parenthesis space equals space sin squared straight x minus cosx comma straight f apostrophe left parenthesis straight x right parenthesis equals 2 sinx. cosx space plus space sinx equals space sin space straight x left parenthesis 2 cosx plus 1 right parenthesis Equating space straight f apostrophe left parenthesis straight x right parenthesis space to space zero. straight f apostrophe left parenthesis straight x right parenthesis space equals space 0 sinx left parenthesis 2 cosx plus 1 right parenthesis space equals space 0 sinx space equals space 0 therefore space straight x space equals space 0 comma space straight pi 2 cosx plus 1 space equals 0 rightwards double arrow cosx space equals space minus 1 half therefore straight x space equals space fraction numerator 5 straight pi over denominator 6 end fraction straight f left parenthesis 0 right parenthesis space equals space sin squared 0 minus cos 0 equals space minus 1 straight f open parentheses fraction numerator 5 straight pi over denominator 6 end fraction close parentheses equals sin squared open parentheses fraction numerator 5 straight pi over denominator 6 end fraction close parentheses minus cos open parentheses fraction numerator 5 straight pi over denominator 6 end fraction close parentheses equals sin squared straight pi over 6 plus cos straight pi over 6 equals 1 fourth minus fraction numerator square root of 3 over denominator 2 end fraction equals open parentheses fraction numerator 1 minus 2 square root of 3 over denominator 4 end fraction close parentheses straight f left parenthesis straight pi right parenthesis equals sin squared straight pi minus cosπ space equals 1

Of these values, the maximum value is 1, and the minimum value is −1.
Thus, the absolute maximum and absolute minimum values of f(x) are 1 and −1, which it attains at x = 0 and

straight x equals straight pi

Some More Questions From Relations and Functions Chapter

Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R’ = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Give an example of a relation which is

(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L₁, L₂) : L₁ is perpendicular to L₂}. Show that R is symmetric but neither reflexive nor transitive.

Determine whether each of the following relations are reflexive, symmetric and transitive :

(i) Relation R in the set A = {1, 2, 3,....., 13, 14} defined as

R = {(x, y) : 3 x – y = 0}

(ii) Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x,y) : y is divisible by x} (iv) Relation R in the set Z of all integers defined as R = {(x,y) : x – y is an integer}

(v) Relation R in the set A of human beings in a town at a particular time given by
(a)    R = {(x, y) : x and y work at the same place}
(b)    R = {(x,y) : x and y live in the same locality}
(c)    R = {(x, y) : x is exactly 7 cm taller than y}
(d)    R = {(x, y) : x is wife of y}
(e)    R = {(x,y) : x is father of y}

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Relations And Functions

Find the absolute maximum and absolute minimum values of the function f given by
$straight f left parenthesis straight x right parenthesis space equals sin squared straight x minus cosx comma space straight x space element of space left parenthesis 0 comma space straight pi right parenthesis$

Some More Questions From Relations and Functions Chapter

Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R’ = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Give an example of a relation which is

(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L₁, L₂) : L₁ is perpendicular to L₂}. Show that R is symmetric but neither reflexive nor transitive.

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.

Check whether the relation R in R defined by R = {(a,b) : a ≤ b³} is refleive, symmetric or transitive.

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For Daily Free Study Material Join wiredfaculty

Relations And Functions

Find the absolute maximum and absolute minimum values of the function f given by

Some More Questions From Relations and Functions Chapter

Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R’ = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Give an example of a relation which is (i) Symmetric but neither reflexive nor transitive.(ii) Transitive but neither reflexive nor symmetric.(iii) Reflexive and symmetric but not transitive.(iv) Reflexive and transitive but not symmetric.(v) Symmetric and transitive but not reflexive.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L1, L2) : L1 is perpendicular to L2}. Show that R is symmetric but neither reflexive nor transitive.

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.

Check whether the relation R in R defined by R = {(a,b) : a ≤ b3} is refleive, symmetric or transitive.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Find the absolute maximum and absolute minimum values of the function f given by
$straight f left parenthesis straight x right parenthesis space equals sin squared straight x minus cosx comma space straight x space element of space left parenthesis 0 comma space straight pi right parenthesis$

Give an example of a relation which is

(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L₁, L₂) : L₁ is perpendicular to L₂}. Show that R is symmetric but neither reflexive nor transitive.

Check whether the relation R in R defined by R = {(a,b) : a ≤ b³} is refleive, symmetric or transitive.