if a+2b+c=4 then find the maximum value of ab+bc+ca
Intuitively, we can locate the maximum by the following reasoning:
Given
a+2b+c=4.....(1)
as a constraint, we look for the maximum of
f(a,b,c)=ab+bc+ca...(2)
First solve for b in terms of a and b from 1 to get:
b=(4-a-c)/2
Substitute in (2) to get
(4c+4a-a²-c²)/2
which is perfectly symmetrical in a and c.
So set a and c each equal to x and find the maximum of
f(x)=8x-2x²
by setting f'(x)=8-4x=0,
we find the maximum of f(x)=f(2)=4 when a=c=2, and b=0.
Alternately, we can find it formally using the Lagrange multiplier method.
Let the objective function
P(a,b,c)=ab+bc+ca+L(a+2b+c-4)
where L is an undetermined constant, and a+2b+c-4=0 is the given constraint.
Differentiating partially with respect to a, b, and c yields the following set of linear equations:
a+2b+c=4 ....(1) given constraint
b+c+L=0 ....(2) ∂P/∂a=0
a+c+2L=0 ...(3) ∂P/∂b=0
b+a+L=0 ...(4) ∂P/∂c=0
By solving the linear system (1) to (4), we get
a=2,b=0,c=2,L=-2 as before.
To find the maximum value of ab + bc + ca, we can use the method of completing the square.
Given the equation a + 2b + c = 4, we can rearrange it as follows: a = 4 - 2b - c
Now, let's rewrite the expression ab + bc + ca using the equation for a:
ab + bc + ca = ab + b(4 - 2b - c) + c(4 - 2b - c)
Expanding and simplifying the expression:
ab + bc + ca = ab + 4b - 2b² - bc + 4c - c²
Rearranging the terms:
ab + bc + ca = -2b² + (a + 4 - c)b + 4c - c²
Now, we can see that this expression can be written in the form of a quadratic equation: ax² + bx + c.
The coefficient of b in our expression is a + 4 - c, which means the maximum value will be achieved when this coefficient is equal to zero.
Therefore, a + 4 - c = 0
a = c - 4
Substituting this value of a back into the original equation:
ab + bc + ca = -2b² + (c - 4 + 4 - c)b + 4c - c²
ab + bc + ca = -2b² + 4b + 4c - c²
To find the maximum value of this quadratic function, we can use the formula x = -b/2a, which gives the x-coordinate of the vertex.
In our case, a = -2, and b = 4.
Substituting these values into the formula:
x = -4 / (2 * -2) = 1
Now, we can substitute x = 1 back into the equation to find the maximum value of ab + bc + ca:
ab + bc + ca = -2(1)² + 4(1) + 4c - c²
ab + bc + ca = -2 + 4 + 4c - c²
ab + bc + ca = 2 + 4c - c²
This expression is now simplified, and to find the maximum value, we need to find the maximum value of 2 + 4c - c².
To find the maximum or minimum of a quadratic function, we can observe the parabola formed by the function. In this case, since the coefficient of the quadratic term is negative (-1), the parabola opens downwards, indicating that it has a maximum value.
The vertex of a parabola is the point (h, k), where h represents the x-coordinate and k represents the y-coordinate.
Since our quadratic function is in the form f(c) = -c² + 4c + 2, the maximum value will occur at the x-coordinate, c = 1, which we obtained earlier.
By substituting c = 1 back into the equation:
ab + bc + ca = 2 + 4(1) - (1)²
ab + bc + ca = 6
Therefore, the maximum value of ab + bc + ca is 6.