Question

020-21
2
For any real numbers a, b, c find the smallest value of the expression 3a + 2762 + 5c? - 18ab - 30c + 237​

Answer 1

Answer:

For any real number a, b, and c, what is the smallest value of the expression 3a^2+27b^2+5c^2-18ab-30c+237?

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Just try to adjust the variables into squares.

The given expression:

3a2+27b2+5c2−18ab−30c+237

Taking 3 common from the coefficients of a2 , b2 and ab terms. Also taking 5 common from the coefficients of c2 and c terms. The expression then becomes:

3(a2+(3b)2–2∗a∗3b)+5(c2–2∗c∗3)+237

The term in the first braces can be condensed into (a−3b)2

The term in the second braces needs a 32 term to be condensed into (c−3)2

So we add 9 inside the second braces and subtract 5∗9 from 237

3(a−3b)2+5(c−3)2+237–45

The final expression becomes:

3(a−3b)2+5(c−3)2+192

The lowest value of this expression occu

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So, minimum value is 192

How can I prove (a+b+c)2≤3a2+3b2+3c2 ?

(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca). Now if we simply multiply the RHS with 3 then RHS has to be greater than (a+b+c)^2 which we can clearly find when LHS is multiplied by 3 then the identity will be balanced. If 6(ab+bc+ca) becomes 0 then 3(a+b+c)^2 will be equal to 3a^2+3b^2+3c^2. Finally if a=b=c=0

If a real number a, b, c, d, & e satisfies a+1=b+2=c+3=d+4=e+5=a+b+c+d+e+3, what is the value of a^2+b^2+c^2+d^2+e^2?