Max Value of $a^2+2a+b^2$ When $2a^2-6a+b^2=0$

  • MHB
  • Thread starter anemone
  • Start date
In summary, the maximum value of the expression $a^2+2a+b^2$ occurs when $a=-\frac{b}{2}$, which is the vertex of the parabola described by the equation $2a^2-6a+b^2=0$. This value can be found by using the formula $a=-\frac{b}{2}$ and then plugging it into the expression. The maximum value can also be visualized as the highest point on the curve of the parabola in the $ab$-plane. The maximum value will always be positive or zero, and it can be affected by changing the values of $a$ and $b$. Increasing $a$ and $
  • #1
anemone
Gold Member
MHB
POTW Director
3,883
115
If the real numbers $a$ and $b$ satisfy the condition $2a^2-6a+b^2=0$, find the maximal value of $a^2+2a+b^2$.
 
Mathematics news on Phys.org
  • #2
anemone said:
If the real numbers $a$ and $b$ satisfy the condition $2a^2-6a+b^2=0$, find the maximal value of $a^2+2a+b^2$.

we need to maximize $a^2+2a+b^2$ or $a^2+2a+b^2-(2a^2-6a+b^2)$ as $2^{nd}$ expression is zero hence constant
or $8a-a^2 = 16-(a^2-8a + 16) = 16-(a-4)^2$
above is maximum when $|a-4|$ minimum.
we need to find it under given constraint
which is
$2a^2 -6a + b^2 = 0$
or $(4a^2 - 12a + 9) + 2b^2 = 9$
or $(2a-3)^2 + b^2 = 9$
or $- 3 < = 2a - 3 <= 3$ or a has to be between 0 and 3
so a = 3 and maximum value of given expression = 15.
 
  • #3
anemone said:
If the real numbers $a$ and $b$ satisfy the condition $2a^2-6a+b^2=0$, find the maximal value of $a^2+2a+b^2$.

My solution:

In order for $a$ to be real, we require:

\(\displaystyle (-6)^2-4(2)(b^2)\ge0\)

\(\displaystyle 9-2b^2\ge0\)

\(\displaystyle 0\le b^2\le\frac{9}{2}\)

In order for $b$ to be real, we require:

\(\displaystyle 6a-2a^2\ge0\)

\(\displaystyle a(3-a)\ge0\implies 0\le a\le3\)

For simplicity, let's write the objective function in terms of $a$ only using the constraint:

\(\displaystyle f(a)=a^2+2a+6a-2a^2=8a-a^2\)

\(\displaystyle f'(a)=8-2a=2(4-a)\)

On the domain for $a$, we find $f$ is strictly increasing, hence:

\(\displaystyle f_{\max}=f(3)=15\)
 
  • #4
Very good job to both of you! And thanks for participating!(Cool)(Wink)
 

FAQ: Max Value of $a^2+2a+b^2$ When $2a^2-6a+b^2=0$

1. What is the maximum value of the expression $a^2+2a+b^2$ when $2a^2-6a+b^2=0$?

The maximum value of the expression $a^2+2a+b^2$ occurs when $a=-\frac{b}{2}$, which is the vertex of the parabola described by the equation $2a^2-6a+b^2=0$. Plugging in this value for $a$, we get the maximum value of $b^2$ as $-\frac{b^2}{4}$. Therefore, the maximum value of the expression is $\frac{3b^2}{4}$.

2. How can I find the value of $a$ and $b$ that will result in the maximum value of the expression $a^2+2a+b^2$?

To find the values of $a$ and $b$ that will result in the maximum value of the expression, we need to find the vertex of the parabola described by the equation $2a^2-6a+b^2=0$. This can be done by using the formula $a=-\frac{b}{2}$ to find the value of $a$ at the vertex, and then plugging this value into the equation to solve for $b$. Once we have the values of $a$ and $b$, we can plug them into the expression $a^2+2a+b^2$ to get the maximum value.

3. Is there a geometric interpretation of the maximum value of the expression $a^2+2a+b^2$ when $2a^2-6a+b^2=0$?

Yes, there is a geometric interpretation of the maximum value. The equation $2a^2-6a+b^2=0$ represents a parabola in the $ab$-plane, and the maximum value of the expression $a^2+2a+b^2$ occurs at the vertex of this parabola. This can be visualized as the highest point on the curve of the parabola.

4. Can the maximum value of the expression $a^2+2a+b^2$ be negative?

No, the maximum value of the expression $a^2+2a+b^2$ cannot be negative. This is because the expression is always the sum of two squared terms, which are always positive. Therefore, the maximum value of the expression will always be positive or zero.

5. How does changing the values of $a$ and $b$ affect the maximum value of the expression $a^2+2a+b^2$ when $2a^2-6a+b^2=0$?

Changing the values of $a$ and $b$ will affect the maximum value of the expression in the following ways:

  • Increasing the value of $a$ will shift the vertex of the parabola to the right, resulting in a higher maximum value for the expression.
  • Decreasing the value of $a$ will shift the vertex of the parabola to the left, resulting in a lower maximum value for the expression.
  • Increasing the value of $b$ will shift the vertex of the parabola upwards, resulting in a higher maximum value for the expression.
  • Decreasing the value of $b$ will shift the vertex of the parabola downwards, resulting in a lower maximum value for the expression.

Similar threads

MHB How Can You Prove the Trigonometric Identity Cos^6A+Sin^6A=1-3Sin^2ACos^2A?
Replies
6
Views
2K
B Complete the square, yes, but what does it mean?
Replies
19
Views
2K
B Is Each Stage of Pascal's Triangle Special?
Replies
11
Views
820
Find the values of a and b to get max/min of certain expression
Replies
21
Views
1K
MHB Integer Challenge: Proving $2A, A+B, C$ integers for $f(x)=Ax^2+Bx+C$
Replies
2
Views
1K
MHB What is the maximum value of $a^2b$?
Replies
10
Views
2K
MHB What is the axis of symmetry of quadratic function f(x)=-2x^2-(1/2)
Replies
2
Views
1K
MHB Is x+1 a Factor of the Polynomial x^3-5x^2+3x+1?
Replies
2
Views
1K
MHB Min Value of $a^2+b^2$ in Quadratic Equation
Replies
1
Views
1K
MHB Prove $b^2-c^2\leq2a^2$ in $\triangle ABC$
Replies
1
Views
937

Hot Threads

Recent Insights

Back
Top