Let f (x) be a polynomial of degree four having extreme values at x =1 an x =2. If $limit as straight x rightwards arrow 0 of open square brackets 1 plus fraction numerator straight f left parenthesis straight x right parenthesis over denominator straight x squared end fraction close square brackets space equals space 3 comma$ then f(2) is equal to

-8

-4

0

4

Solution

Any function have extreme values (maximum and minimum) at its critical points, where f'(x)= 0
Since, the function have extreme values at x =1 and x=2
therefore, f'(x) = 0 at x =1 and x= 2
⇒ f'(1) = 0 and f'(2) = 0
Also, it is given that
$straight l im with straight x rightwards arrow 0 below space open square brackets 1 plus fraction numerator straight f left parenthesis straight x right parenthesis over denominator straight x squared end fraction close square brackets space equals space 3 rightwards double arrow 1 plus limit as straight x space rightwards arrow 0 of fraction numerator straight f left parenthesis straight x right parenthesis over denominator straight x squared end fraction space equals space 3 rightwards double arrow space limit as straight x rightwards arrow 0 of space fraction numerator straight f left parenthesis straight x right parenthesis over denominator straight x squared end fraction space equals space 2$
⇒ f(x) will be of the form
ax⁴ + bx³ + 2x⁴
f(x) is four degree polynomial]
Let f(x) = ax4 +bx3 +2x2
⇒ f'(x) = 4ax3 + 3bx2+ 4x
⇒ f'(1) = 4a +3b+4 = 0
and f'(2) 32a + 12b +8 = 0
⇒ 8a + 3b + 2 = 0
On solving Eqs. (i) and (ii), we get
a = 1/2, b = -2
$straight f left parenthesis straight x right parenthesis space space equals space straight x to the power of 4 over 2 space minus space 2 straight x cubed space plus 2 straight x squared$
f(2) = 8 - 16 +8 = 0

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

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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Relations And Functions

Let f (x) be a polynomial of degree four having extreme values at x =1 an x =2. If then f(2) is equal to -8 -4 0 4

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

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.