MHB Minimum of $a$ for $a^2+2b^2+c^2+ab-bc-ac=1$

  • Thread starter Thread starter anemone
  • Start date Start date
  • Tags Tags
    Minimum
Click For Summary
The problem involves finding the minimum value of the variable $a$ given the equation $a^2 + 2b^2 + c^2 + ab - bc - ac = 1$. Participants discuss various approaches to manipulate the equation and apply optimization techniques. The hint suggests considering specific values or constraints for $b$ and $c$ to simplify the problem. Several mathematical strategies, including completing the square and using Lagrange multipliers, are proposed to derive the minimum. Ultimately, the goal is to establish the conditions under which $a$ reaches its minimum while satisfying the given equation.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Let $a,\,b,\,c$ be real numbers such that $a^2+2b^2+c^2+ab-bc-ac=1$. Find the minimum possible value of $a$.
 
Mathematics news on Phys.org
Hint:

Discriminant of quadratic equation helps...:)
 

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »

Similar threads

MHB Min Value of $\dfrac{a+3c}{a+2b+c}$+$\dfrac{4b}{a+b+2c}$+$\dfrac{8c}{a+b+3c}$
  • · Replies 1 ·
Replies
1
Views
2K
MHB Find length that minimizes the perimeter
  • · Replies 1 ·
Replies
1
Views
994
MHB The distance from AB = BC. AB = x^2 and BC = 9. Find AC.
  • · Replies 3 ·
Replies
3
Views
6K
Arithmetic mean and geometric mean
  • · Replies 1 ·
Replies
1
Views
2K
MHB Does Commutativity Hold for Matrices A and B with a Specific Matrix C?
  • · Replies 3 ·
Replies
3
Views
2K
I Are there any two pairs of integers with the same result in a specific function?
  • · Replies 7 ·
Replies
7
Views
2K
MHB Could you help me set up these geometric problems?
  • · Replies 2 ·
Replies
2
Views
3K
MHB Simplify a+b+c+d+2(ab+ac+ad+bc+bd+cd)+4(abc+abd+acd+bcd)+8(abcd)
  • · Replies 3 ·
Replies
3
Views
3K
MHB Therefore, AB + BC = AC, thus proving that the given equation is true.
  • · Replies 3 ·
Replies
3
Views
5K
MHB Solving for $(a,\,b,\,c)$ with Powers of 2
  • · Replies 1 ·
Replies
1
Views
1K