1800-150-456-789
FlagBACK
Flag Algebra> A.P...
question mark

a+b+c = 3,then find the greatest value of a^2b^3c^2.

kartik kumar , 10 Years ago
Grade 12
anser 2 Answers
Nishant Vora

Last Activity: 10 Years ago

Hello Student, Please find the solution

Given a+b+c =3
Write this as
\frac{a}{2} + \frac{a}{2}+ \frac{b}{3}+ \frac{b}{3}+ \frac{b}{3}+ \frac{c}{2}+ \frac{c}{2} = 3

Now we will use AM >= GM

So
\frac{[\frac{a}{2} + \frac{a}{2}+ \frac{b}{3}+ \frac{b}{3}+ \frac{b}{3}+ \frac{c}{2}+ \frac{c}{2} ]}{7}} >= [\frac{a}{2}*\frac{a}{2}*\frac{b}{3}*\frac{b}{3}*\frac{b}{3}*\frac{c}{2}*\frac{c}{2}]^\frac{1}{7}

Therefore,
\frac{3}{7} >= [\frac{a}{2}*\frac{a}{2}*\frac{b}{3}*\frac{b}{3}*\frac{b}{3}*\frac{c}{2}*\frac{c}{2}]^\frac{1}{7}
[\frac{3}{7}]^7 >= [\frac{a}{2}*\frac{a}{2}*\frac{b}{3}*\frac{b}{3}*\frac{b}{3}*\frac{c}{2}*\frac{c}{2}]
[\frac{3}{7}]^7 >= \frac{a^2b^3c^2}{2^23^32^2}
[\frac{3}{7}]^7*2^23^32^2 >=a^2b^3c^2
So greatest value of a^2b^3c^2 is
\frac{3^{10}2^4}{7^7}

ankit singh

Last Activity: 4 Years ago

iven: a + 2b + 3c = 4

=> a = 4- 2b - 3c

To find : Minimum value of a² + b² + c² OR to find min value of (4–2b - 3c)² + b² + c² , here we have 2 variables related by the above shown expression. Let's suppose its minimum value is ‘m'

m = (4–2b-3c)² + b² + c² . . . . . . . . . . . (1)

The derivative of m wrt b = Polynomial can be differentiated by power rule (d/dx) x^n = n x^(n-1)

dm/ db = (-2) (2) ( 4–2b - 3c)^(2–1) + 2b=0 ( by quotient rule)

=> -16 + 8b +12c + 2b = 0

=> 10b + 12c = 16

=> 5b + 6c = 8 . . . . . . . . . (2)

Next, the derivative of m wrt c =

dm/dc = (-3) (2) ( 4–2b - 3c)^(2–1) + 2c =0

=> -24 + +12b +18c + 2c = 0

=> 12b + 20c = 24

=> 3b + 5c = 6 . . . . . . . . . . . .(3)

By solving eq (1) & eq(2)

We get b= 4/7 , c = 6/7

Since, ‘m’ is the minimum value of the expression

So, we substitute the values of b & c in eq (2) & (3), to get the value of m.

(4 - 2*4/7 - 3*6/7 )² + (4/7)² + (6/7)²

= ( 4- 8/7 - 18/7)² + 16/49 + 36/49

= 4/49 + 16/49 + 36/49

= 56/49

= 8/7

 

Provide a better Answer & Earn Cool Goodies

star
LIVE ONLINE CLASSES

Prepraring for the competition made easy just by live online class.

tv

Full Live Access

material

Study Material

removal

Live Doubts Solving

assignment

Daily Class Assignments


Ask a Doubt

Get your questions answered by the expert for free

Other Related Questions on algebra

question mark
What is a binomial expression?
algebra
1 Answer Available

Last Activity: 1 Year ago

question mark
Can we divide two vectors? If this is possible then how?
algebra
1 Answer Available

Last Activity: 1 Year ago

question mark
How do you write “the product of 18 and q” as an algebraic expression?
algebra
1 Answer Available

Last Activity: 1 Year ago

question mark
दहावीच्या आय सी एस इ विद्यार्थ्यांनी पुढे काय करायला पाहिजे
algebra
0 Answer Available

Last Activity: 2 Years ago

question mark
Find the least number of square tiles by which the floor of a room of dimensions 16.58 m and 8.32 m can be covered completely. why HCF and not LCM as “least” keywor dis given?
algebra
1 Answer Available

Last Activity: 2 Years ago