Problem 16
Question
If \(a, b, c \in R^{+}\), then the minimum value of \(a\left(b^{2}+c^{2}\right)+b\left(c^{2}+a^{2}\right)\) \(+c\left(a^{2}+b^{2}\right)\) is equal to a. \(a b c\) b. \(2 a b c\) c. \(3 a b c\) d. \(6 a b c\)
Step-by-Step Solution
Verified Answer
The minimum value is option d: \(6abc\).
1Step 1: Analyze the Expression
The given expression is \(a(b^2+c^2) + b(c^2+a^2) + c(a^2+b^2)\). We want to find the minimum value of this expression for positive real numbers \(a, b, c\).
2Step 2: Apply Simplification
Rewrite the expression as \(ab^2 + ac^2 + bc^2 + ba^2 + ca^2 + cb^2\). Notice that this expression is symmetric in \(a, b, c\).
3Step 3: Use Symmetry and Substitution
To find the minimum, let's assume symmetry conditions like \(a = b = c\). Substitute \(a = b = c = x\) into the expression, turning it into \(3x(x^2 + x^2) = 6x^3\).
4Step 4: Calculate with the Assumption
Substitute \(x = a = b = c\) into the expression, leading to the minimum being \(6a^3\). Since \(x = a = b = c\), this is equivalent to \(6abc\) in terms of the original variables.
5Step 5: Conclusion and Selection
The calculated minimum value of the expression is \(6abc\), which matches option d.
Key Concepts
Symmetry in AlgebraExpression SimplificationAlgebraic Manipulation
Symmetry in Algebra
Symmetry in algebra plays a crucial role in simplifying complex expressions. It allows us to uncover patterns and use them to make algebraic calculations more manageable. When an expression is symmetric in terms of its variables, it means that any permutation of those variables will leave the expression unchanged.
In the context of the exercise, the expression \(a(b^2+c^2) + b(c^2+a^2) + c(a^2+b^2)\) is symmetric in the variables \(a, b,\) and \(c\). This symmetry suggests that the values of \(a, b,\) and \(c\) can be interchanged without affecting the expression.
Leveraging this symmetry, we simplify the process by assuming a condition where all variables are equal, such as \(a = b = c = x\). This reduces the complexity of finding the minimum value because it narrows down the problem to solving for one variable. Recognizing symmetry helps identify these simplifications and can be a powerful tool in algebraic manipulation.
In the context of the exercise, the expression \(a(b^2+c^2) + b(c^2+a^2) + c(a^2+b^2)\) is symmetric in the variables \(a, b,\) and \(c\). This symmetry suggests that the values of \(a, b,\) and \(c\) can be interchanged without affecting the expression.
Leveraging this symmetry, we simplify the process by assuming a condition where all variables are equal, such as \(a = b = c = x\). This reduces the complexity of finding the minimum value because it narrows down the problem to solving for one variable. Recognizing symmetry helps identify these simplifications and can be a powerful tool in algebraic manipulation.
Expression Simplification
Expression simplification entails reducing an expression to its most compact form, making algebraic manipulation more straightforward and less error-prone. Simplification is done by combining like terms, factoring, or using algebraic identities.
In our original problem, the expression was initially \(a(b^2+c^2) + b(c^2+a^2) + c(a^2+b^2)\). By expanding it, we got \(ab^2 + ac^2 + bc^2 + ba^2 + ca^2 + cb^2\), a step known as distributing terms.
The main aim here was to recognize the symmetry for further simplifications. Once expanded, the expression can be further evaluated under uniform conditions, like assuming \(a = b = c\) for simplicity, helping achieve the insight necessary to find minimum or specific critical values.
In our original problem, the expression was initially \(a(b^2+c^2) + b(c^2+a^2) + c(a^2+b^2)\). By expanding it, we got \(ab^2 + ac^2 + bc^2 + ba^2 + ca^2 + cb^2\), a step known as distributing terms.
The main aim here was to recognize the symmetry for further simplifications. Once expanded, the expression can be further evaluated under uniform conditions, like assuming \(a = b = c\) for simplicity, helping achieve the insight necessary to find minimum or specific critical values.
Algebraic Manipulation
Algebraic manipulation refers to the various operations we can perform on algebraic expressions to achieve desired goals, like solving equations or finding minimum values. It relies on using basic algebraic operations such as addition, subtraction, multiplication, division, as well as advanced techniques like factoring or expanding.
In this exercise, manipulation was used to find the minimum value by assuming all variables equal, substituting \(a = b = c = x\). This transformed the original expression into a simpler form, \(3x(x^2 + x^2) = 6x^3\), revealing that its minimum value is \(6abc\) when \(x = a = b = c\).
Through strategic substitutions and leveraging known identities or properties like symmetry, algebraic manipulation becomes a potent tool in breaking down and conquering seemingly complex expressions, thereby finding simpler means to reach conclusions.
In this exercise, manipulation was used to find the minimum value by assuming all variables equal, substituting \(a = b = c = x\). This transformed the original expression into a simpler form, \(3x(x^2 + x^2) = 6x^3\), revealing that its minimum value is \(6abc\) when \(x = a = b = c\).
Through strategic substitutions and leveraging known identities or properties like symmetry, algebraic manipulation becomes a potent tool in breaking down and conquering seemingly complex expressions, thereby finding simpler means to reach conclusions.
Other exercises in this chapter
Problem 14
If \(a, b, c \in R^{+}\), then \(\frac{b c}{b+c}+\frac{a c}{a+c}+\frac{a b}{a+b}\) is always a. \(\leq \frac{1}{2}(a+b+c)\) b. \(\geq \frac{1}{3} \sqrt{a b c}\)
View solution Problem 15
If \(a, b, c \in R^{*}\), then \((a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\) is always a. \(\geq 12\) b. \(\geq 9\) c. \(\leq 12\) d. none of thes
View solution Problem 17
If \(a, b, c, d \in R^{+}\)and \(a, b, c, d\) are in H.P., then a. \(a+d>b+c\) b. \(a+b>c+d\) c. \(a+c>b+d\) d. none of these
View solution Problem 18
If \(a, b, c \in R^{+}\)such that \(a+b+c=18\), then the maximum value of \(a^{2} b^{3} c^{4}\) is equad to a. \(2^{18} \times 3^{2}\) b. \(2^{18} \times 3^{3}\
View solution