Home
Class 12
MATHS
Let a ,ba n dc be real numbers such that...

Let `a ,ba n dc` be real numbers such that `a+2b+c=4` . Find the maximum value of `(a b+b c+c a)dot`

Text Solution

Verified by Experts

Given,
`a + 2b + c = 4 or a = 4 - 2b -c`
Let `ab + bc + ca = x or a(b +c) + be = x`
or `(4 - 2b - c) (b + c) + bc = x`
or `4b + 4c - 2b^(2) - 2bc - bc - c^(2) + be = x `
or `2b^(2) - 4b + 2bc - 4c + c^(2) + x = 0`
or ` 2b^(2) + 2(c - 2) b - 4c + c^(2) + x = 0`
Since b `in` R, so
` 4(c - 2)^(2) - 4xx2(-4c + c^(2) + x) ge 0`
or `c^(2) - 4c + 4 + 8c - 2c^(2) - 2x ge 0`
or `c^(2) - 4c + 2x - 4 le 0`
Since c `in` R, so
`16 - 4 (2x - 4) ge 0 rArr x le 4 `
`therefore` max ( ab + bc + ac )= 4`
Promotional Banner

Topper's Solved these Questions

  • THEORY OF EQUATIONS

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 2.1|3 Videos

  • THEORY OF EQUATIONS

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 2.2|5 Videos

  • THEORY OF EQUATIONS

    CENGAGE ENGLISH|Exercise ILLUSTRATION|122 Videos

  • STRAIGHT LINES

    CENGAGE ENGLISH|Exercise ARCHIVES (NUMERICAL VALUE TYPE)|1 Videos

  • THREE DIMENSIONAL GEOMETRY

    CENGAGE ENGLISH|Exercise All Questions|294 Videos

Similar Questions

Explore conceptually related problems

Let a ,b and c be real numbers such that a+2b+c=4 . Find the maximum value of (a b+b c+c a)dot

If a,b,c are positive real numbers such that a + b +c=18, find the maximum value of a^2b^3c^4

Let a, b, c , d he real numbers such that a + b+c+d = 10 , then the minimum value of a^2 cot 9^@+b^2 cot 27^@+c^2 cot 63^@+d^2 cot 81^@ is sqrtn ,(n in N) find n.

Let a ,b ,and c be real numbers such that 4a+2b+c=0 and a b > 0. Then the equation a x^2+b x+c = 0

Let a, b and c be real numbers such that a - 7b + 8c = 4 and 8a + 4b - c = 7. What is the value of a^(2) - b^(2) + c^(2) .

Let a, b and c be real numbers such that 4a + 2b + c = 0 and ab gt 0. Then the equation ax^(2) + bx + c = 0 has

Let a,b,c and d be non-negative real number such that a ^(5)+b^(5) le 1 and c ^(5) + d^(5) le 1. Find the maximum value of a ^(2) c ^(3) +b^(2) d ^(3).

Let a,b,c,d be real numbers such that |a-b|=2, |b-c|=3, |c-d|=4 Then the sum of all possible values of |a-d|=

a ,b ,c are three complex numbers on the unit circle |z|=1, such that a b c=a+b+cdot Then find the value of |a b+b c+c a|dot

a ,b ,c are three complex numbers on the unit circle |z|=1, such that a b c=a+b+c dot Then find the value of |a b+b c+c a|dot